Acoustic wave device

ABSTRACT

An acoustic wave device includes a silicon substrate, a polysilicon layer provided on the silicon substrate, a silicon oxide layer directly or indirectly provided on the polysilicon layer, a piezoelectric layer directly or indirectly provided on the silicon oxide layer, and an interdigital transducer electrode provided on the piezoelectric layer. A plane orientation of the silicon substrate is any one of (100), (110), and (111), and, where a wave length that is defined by an electrode finger pitch of the interdigital transducer electrode is λ, a thickness of the piezoelectric layer is less than or equal to about 1λ.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority to Japanese PatentApplication No. 2020-169095 filed on Oct. 6, 2020 and is a ContinuationApplication of PCT Application No. PCT/JP2021/035803 filed on Sep. 29,2021. The entire contents of each application are hereby incorporatedherein by reference.

BACKGROUND OF THE INVENTION 1. Field of the Invention

The present invention relates to an acoustic wave device.

2. Description of the Related Art

Hitherto, acoustic wave devices are widely used in filters of mobilephones, and the like. International Publication No. 2018/151147describes an example of the acoustic wave devices. In the acoustic wavedevice, an interdigital transducer electrode is provided on a multilayersubstrate. In the multilayer substrate, a silicon substrate, a siliconoxide layer, a silicon nitride layer, and a piezoelectric substrate arelaminated in this order. The plane orientation of the silicon substrateis set to (100), (110), or (111). Thus, a bulk wave spurious is reducedor prevented.

SUMMARY OF THE INVENTION

However, even when the plane orientation of the silicon substrate is setto (100), (110), or (111) as in the case of the acoustic wave devicedescribed in International Publication No. 2018/151147, it is difficultto sufficiently reduce ripple caused by higher-order modes.

Preferred embodiments of the present invention provide acoustic wavedevices each capable of reducing higher-order modes in a wide band.

An acoustic wave device according to a preferred embodiment of thepresent invention includes a silicon substrate, a polysilicon layerprovided on the silicon substrate, a silicon oxide layer directly orindirectly provided on the polysilicon layer, a piezoelectric layerdirectly or indirectly provided on the silicon oxide layer, and aninterdigital transducer electrode provided on the piezoelectric layer. Aplane orientation of the silicon substrate is any one of (100), (110),and (111). Where a wave length that is defined by an electrode fingerpitch of the interdigital transducer electrode is λ, a thickness of thepiezoelectric layer is less than or equal to about 1λ.

With the acoustic wave devices according to preferred embodiments of thepresent invention, higher-order modes are reduced in a wide band.

The above and other elements, features, steps, characteristics andadvantages of the present invention will become more apparent from thefollowing detailed description of the preferred embodiments withreference to the attached drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an elevational cross-sectional view of a portion of anacoustic wave device according to a first preferred embodiment of thepresent invention.

FIG. 2 is a plan view of the acoustic wave device according to the firstpreferred embodiment of the present invention.

FIG. 3 is a schematic diagram that shows the definition of thecrystallographic axes of silicon.

FIG. 4 is a schematic diagram that shows a (111) plane of silicon.

FIG. 5 is a diagram when the crystallographic axes of the (111) plane ofsilicon are viewed from an XY-plane in the first preferred embodiment ofthe present invention.

FIG. 6 is a schematic diagram that shows a (100) plane of silicon.

FIG. 7 is a schematic diagram that shows a (110) plane of silicon.

FIG. 8 is a graph that shows the phase characteristics of the acousticwave device according to the first preferred embodiment of the presentinvention and the phase characteristics of an acoustic wave deviceaccording to a first comparative example.

FIG. 9 is a graph that shows the impedance frequency characteristics ofan acoustic wave device according to a first reference example.

FIG. 10 is a graph that shows the impedance frequency characteristics ofan acoustic wave device according to a second reference example.

FIG. 11 is a graph that shows the impedance frequency characteristics ofthe acoustic wave device according to the first preferred embodiment ofthe present invention.

FIG. 12 is a schematic cross-sectional view for illustrating adirectional vector k₁₁₁.

FIG. 13 is a schematic plan view for illustrating the directional vectork₁₁₁.

FIG. 14 is a schematic diagram that shows a [11-2] direction of silicon.

FIG. 15 is a schematic diagram for illustrating an angle α₁₁₁.

FIG. 16 is a graph that shows the phase characteristics of the acousticwave device of which α₁₁₁ is 60° in the first preferred embodiment ofthe present invention.

FIG. 17 is an elevational cross-sectional view around a pair ofelectrode fingers of an acoustic wave device according to a secondpreferred embodiment of the present invention.

FIG. 18 is a graph that shows the phase characteristics of the acousticwave device according to the second preferred embodiment of the presentinvention.

FIG. 19 is an elevational cross-sectional view around a pair ofelectrode fingers of an acoustic wave device according to a thirdpreferred embodiment of the present invention.

FIG. 20 is a graph that shows the phase characteristics of the acousticwave device according to the third preferred embodiment of the presentinvention.

FIG. 21 is an elevational cross-sectional view around a pair ofelectrode fingers of an acoustic wave device according to a fourthpreferred embodiment of the present invention.

FIG. 22 is a graph that shows the phase characteristics of the acousticwave device according to the fourth preferred embodiment of the presentinvention.

FIG. 23 is a graph that shows the phase characteristics of an acousticwave device according to a fifth preferred embodiment of the presentinvention and the phase characteristics of an acoustic wave deviceaccording to a second comparative example.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Hereinafter, the present invention will be clarified by describingspecific preferred embodiments of the present invention with referenceto the drawings.

It should be noted that each of the preferred embodiments described inthe specification is illustrative and that partial replacements orcombinations of components are possible among different preferredembodiments.

FIG. 1 is an elevational cross-sectional view of a portion of anacoustic wave device according to a first preferred embodiment of thepresent invention. FIG. 2 is a plan view of the acoustic wave deviceaccording to the first preferred embodiment. FIG. 1 is a cross-sectionalview taken along the line I-I in FIG. 2 .

As shown in FIG. 1 , the acoustic wave device 1 includes a multilayersubstrate 9. The multilayer substrate 9 includes a silicon substrate 2,a polysilicon layer 3, a silicon oxide layer 5, and a piezoelectriclayer 7. More specifically, the polysilicon layer 3 is provided on thesilicon substrate 2. The silicon oxide layer 5 is provided on thepolysilicon layer 3. The piezoelectric layer 7 is provided on thesilicon oxide layer 5.

In the present preferred embodiment, the silicon substrate 2 is asilicon monocrystal substrate. The plane orientation of the siliconsubstrate 2 is (111). Of course, the plane orientation of the siliconsubstrate 2 may be any one of (100), (110), and (111).

Silicon oxides in the silicon oxide layer 5 are expressed by SiO_(x). xis a positive number. In the present preferred embodiment, the siliconoxide layer 5 is an SiO₂ layer. Of course, x is not limited to two.

In the present preferred embodiment, the piezoelectric layer 7 is alithium tantalate layer with a cut angle of 40°. Of course, the cutangle and material of the piezoelectric layer 7 are not limited thereto.For example, lithium niobate may be used as the material of thepiezoelectric layer 7.

The interdigital transducer electrode 8 is provided on the piezoelectriclayer 7. Acoustic waves are excited by applying an alternating-currentvoltage to the interdigital transducer electrode 8. As shown in FIG. 2 ,a reflector 14 and a reflector 15 are provided on the piezoelectriclayer 7 respectively on both sides of the interdigital transducerelectrode 8 in an acoustic wave propagation direction. In this way, theacoustic wave device 1 according to the present preferred embodiment isa surface acoustic wave resonator. Although not limited thereto, theacoustic wave device according to the present invention may be a filterdevice, a multiplexer, or the like, that includes a plurality of surfaceacoustic wave resonators.

As shown in FIG. 2 , the interdigital transducer electrode 8 has a firstbusbar 16, a second busbar 17, a plurality of first electrode fingers18, and a plurality of second electrode fingers 19. The first busbar 16and the second busbar 17 are opposite to each other. One ends of thefirst electrode fingers 18 are connected to the first busbar 16. Oneends of the second electrode fingers 19 are connected to the secondbusbar 17. The plurality of first electrode fingers 18 and the pluralityof second electrode fingers 19 interdigitate with each other. In thespecification, the acoustic wave propagation direction is assumed as anX direction. A direction in which the first electrode fingers 18 and thesecond electrode fingers 19 of the interdigital transducer electrode 8extend is assumed as a Y direction. A thickness direction of each of theinterdigital transducer electrode 8, the piezoelectric layer 7, thesilicon substrate 2, and the like is assumed as a Z direction.

The interdigital transducer electrode 8 includes a multilayer metalfilm. More specifically, in the multilayer metal film, a Ti layer, anAlCu layer, and a Ti layer are laminated in this order. The reflector 14and the reflector 15 are also made of a material similar to that of theinterdigital transducer electrode 8. Of course, the materials of theinterdigital transducer electrode 8, the reflector 14, and the reflector15 are not limited thereto. Alternatively, the interdigital transducerelectrode 8, the reflector 14, and the reflector 15 may be made up of asingle-layer metal film.

Where a wave length that is defined by the electrode finger pitch of theinterdigital transducer electrode 8 is A, the thickness of thepiezoelectric layer 7 is less than or equal to about 1λ, for example.The electrode finger pitch is an electrode finger center-to-centerdistance between adjacent electrode fingers. The electrode finger pitchis, specifically, a distance connecting central points of adjacentelectrode fingers in the acoustic wave propagation direction, that is,the X direction. When the electrode finger center-to-center distance isnot constant, the electrode finger pitch is assumed as an average valueof the electrode finger center-to-center distance.

A protective film may be provided on the piezoelectric layer 7 so as tocover the interdigital transducer electrode 8. In this case, theinterdigital transducer electrode 8 is less likely to break. Anappropriate dielectric may be used as the protective film. When, forexample, a silicon oxide is used as the protective film,frequency-temperature characteristics (TCF) are increased. When asilicon nitride is used as the protective film, frequency is easilyadjusted by adjusting the thickness of the protective film. Of course,the protective film does not need to be provided.

Some of the unique features of the present preferred embodiment arethat, in the multilayer substrate 9, the silicon substrate 2, thepolysilicon layer 3, the silicon oxide layer 5, and the piezoelectriclayer 7 are laminated, and the thickness of the piezoelectric layer 7 isless than or equal to about 1λ, for example. Thus, higher-order modesare reduced in a wide band. The details of the advantageous effects willbe described below together with the definition and the like ofcrystallographic axes and plane orientation.

FIG. 3 is a schematic diagram that shows the definition of thecrystallographic axes of silicon. FIG. 4 is a schematic diagram thatshows a (111) plane of silicon. FIG. 5 is a diagram when thecrystallographic axes of the (111) plane of silicon is viewed from anXY-plane in the first preferred embodiment. FIG. 6 is a schematicdiagram that shows a (100) plane of silicon. FIG. 7 is a schematicdiagram that shows a (110) plane of silicon.

As shown in FIG. 3 , a silicon monocrystal has a diamond structure. Inthe specification, the crystallographic axes of silicon that is acomponent of the silicon substrate 2 are assumed as [X_(Si), Y_(Si),Z_(Si)]. As for silicon, an X_(Si)-axis, a Y_(Si)-axis, and aZ_(Si)-axis are equivalent to one another due to the symmetry of acrystal structure. As shown in FIG. 5 , there is a three-fold symmetryin the (111) plane, and an equivalent crystal structure is obtained by120° rotation.

As described above, the plane orientation of the silicon substrate 2according to the present preferred embodiment is (111). The fact thatthe plane orientation is (111) means that a substrate or a layer is cutalong the (111) plane orthogonal to the crystallographic axesrepresented by Miller indices [111] in the crystal structure of siliconhaving a diamond structure. The (111) plane is a plane shown in FIGS. 4and 5 . Of course, the (111) plane includes other crystallographicallyequivalent planes.

On one hand, the fact that the plane orientation is (100) means that asubstrate or a layer is cut along the (100) plane orthogonal to thecrystallographic axes represented by Miller indices [100] in the crystalstructure of silicon having a diamond structure. There is a four-foldsymmetry in the (100) plane, and an equivalent crystal structure isobtained by 90° rotation. The (100) plane is a plane shown in FIG. 6 .

On the other hand, the fact that the plane orientation is (110) meansthat a substrate or a layer is cut along the (110) plane orthogonal tothe crystallographic axes represented by Miller indices [110] in thecrystal structure of silicon having a diamond structure. There is atwo-fold symmetry in the (110) plane, and an equivalent crystalstructure is obtained by 180° rotation. The (110) plane is a plane shownin FIG. 7 .

Here, the present preferred embodiment and a first comparative exampleare compared with each other to demonstrate that higher-order modes arereduced in a wide band according to the present preferred embodiment.The first comparative example differs from the present preferredembodiment in that, in the multilayer substrate, a silicon nitride layeris laminated instead of the polysilicon layer.

FIG. 8 is a graph that shows the phase characteristics of the acousticwave device according to the first preferred embodiment and the phasecharacteristics of an acoustic wave device according to the firstcomparative example.

As indicated by the arrow A in FIG. 8 , in each of the first preferredembodiment and the first comparative example, a higher-order mode around1.5 times the resonant frequency is reduced. However, in the firstcomparative example, it appears that, as indicated by the arrow B, alarge spurious due to a higher-order mode is generated around 2.2 timesthe resonant frequency. In contrast, in the first preferred embodiment,it appears that a higher-order mode is effectively reduced around thefrequency indicated by the arrow B. More specifically, in the firstpreferred embodiment, a higher-order mode is reduced to less than−80[deg.] at not only frequencies around 1.5 times the resonantfrequency but also frequencies around 2.2 times the resonant frequency.

The reason will be described by using a first reference example and asecond reference example. In the first reference example, a multilayersubstrate is a multilayer body of a silicon substrate, a silicon oxidelayer, and a piezoelectric layer. The plane orientation of the siliconsubstrate in the first reference example is (111). In the secondreference example, a multilayer substrate is a multilayer body of apolysilicon substrate, a silicon oxide layer, and a piezoelectric layer.The piezoelectric layer according to the first reference example and thepiezoelectric layer according to the second reference example arelithium tantalate layers.

FIG. 9 is a graph that shows the impedance frequency characteristics ofan acoustic wave device according to the first reference example. FIG.10 is a graph that shows the impedance frequency characteristics of anacoustic wave device according to the second reference example. FIG. 11is a graph that shows the impedance frequency characteristics of theacoustic wave device according to the first preferred embodiment.

As indicated by the arrow A in FIG. 9 , a higher-order mode around 1.5times the resonant frequency is reduced in the first reference example.This is due to the fact that the plane orientation of the siliconsubstrate is (111). However, as indicated by the arrow B, a higher-ordermode around 2.2 times the resonant frequency is not sufficiently reducedin the first reference example.

On the other hand, as indicated by the arrow B in FIG. 10 , ahigher-order mode around 2.2 times the resonant frequency is reduced inthe second reference example. This is due to the fact that thehigher-order mode is made as a leaky mode because the polysiliconsubstrate is provided. However, as indicated by the arrow A, ahigher-order mode around 1.5 times the resonant frequency is notsufficiently reduced.

In contrast, in the first preferred embodiment, the multilayer substrate9 includes both the silicon substrate 2 and the polysilicon layer 3, andthe plane orientation of the silicon substrate 2 is (111). In addition,the thickness of the piezoelectric layer 7 is less than or equal toabout 1λ. With this configuration, as shown in FIG. 11 , both ahigher-order mode around 1.5 times the resonant frequency and ahigher-order mode around 2.2 times the resonant frequency areeffectively reduced. In this way, in the first preferred embodiment,higher-order modes are reduced in a wide band.

Incidentally, in the first preferred embodiment, the polysilicon layer 3and the silicon oxide layer 5 are provided between the silicon substrate2 and the piezoelectric layer 7. The inventor of the present applicationdiscovered that, even in such a case, higher-order modes were furtherreliably reduced by defining the relationship between the planeorientation of the silicon substrate 2 and the crystallographic axes ofthe piezoelectric layer 7. Here, a directional vector that is defined inaccordance with the direction of crystallographic axes of thepiezoelectric layer 7 is assumed as k. An angle that is defined inaccordance with the relationship between the crystallographic axes ofthe piezoelectric layer 7 and the plane orientation of the siliconsubstrate 2 is assumed as α. The directional vector k is any one ofthree vectors k₁₁₁, k₁₁₀, and k₁₀₀. The angle α is any one of threeangles α₁₁₁, α₁₁₀, and α₁₀₀. k₁₁₁ and α₁₁₁, k₁₁₀ and α₁₁₀, and k₁₀₀ andα₁₀₀ respectively correspond to the plane orientations (111), (110), and(100). Hereinafter, the details of the directional vector k and theangle α will be described.

FIG. 12 is a schematic cross-sectional view for illustrating thedirectional vector kill. FIG. 13 is a schematic plan view forillustrating the directional vector kill. The plane orientation of thesilicon substrate 2 in FIG. 12 is (111).

FIGS. 12 and 13 show an example of a case where the Euler angles of thepiezoelectric layer 7 are (0°, −35°, 0°). The (111) plane of the siliconsubstrate 2 is in contact with the piezoelectric layer 7.

Here, as shown in FIG. 12 , a directional vector obtained by projectingthe Z_(P)-axis of a piezoelectric body LiTaO₃ that is a component of thepiezoelectric layer 7 onto the (111) plane of the silicon substrate 2 isassumed as km. As shown in FIGS. 12 and 13 , the directional vector kmis parallel to the Y direction that is a direction in which theelectrode fingers of the interdigital transducer electrode 8 extend.

FIG. 14 is a schematic diagram that shows a [11-2] direction of silicon.FIG. 15 is a schematic diagram for illustrating the angle α₁₁₁.

As shown in FIG. 14 , the [11-2] direction of silicon is represented asa resultant vector of a unit vector in an X_(Si) direction, a unitvector in a Y_(Si) direction, and a vector minus twice a unit vector ina Z_(Si) direction in the crystal structure of silicon. As shown in FIG.15 , the angle α₁₁₁ is an angle between the directional vector km andthe [11-2] direction of silicon that is a component of the siliconsubstrate 2. As described above, from the symmetry of the crystal ofsilicon, the [11-2] direction, a [1-21] direction, and a [-211]direction are equivalent.

On one hand, in a silicon substrate of which the plane orientation is(110), a directional vector obtained by projecting the Z_(P)-axis ontothe (110) plane of the silicon substrate is assumed as kilo. The angleα₁₁₀ is an angle between the directional vector kilo and the [001]direction of silicon that is a component of the silicon substrate. Fromthe symmetry of silicon, the [001] direction, a [100] direction, and a[010] direction are equivalent.

On the other hand, in a silicon substrate of which the plane orientationis (100), a directional vector obtained by projecting the Z_(P)-axisonto the (100) plane of the silicon substrate is assumed as k₁₀₀. Theangle α₁₀₀ is an angle between the directional vector k₁₀₀ and the [001]direction of silicon that is a component of the silicon substrate.

Regardless of whether the silicon substrate is laminated directly on thepiezoelectric layer or laminated indirectly on the piezoelectric layerwith another layer interposed therebetween, the definition of thedirectional vector k and the angle α is the same.

As described above, the plane orientation of the silicon substrate 2 isnot limited to (111). The plane orientation of the silicon substrate 2may be any one of (100), (110), and (111). The angle α is any one ofthree angles, that is, the angle α₁₀₀, the angle α₁₁₀, and the angleα₁₁₁. More specifically, when the plane orientation of the siliconsubstrate 2 is (100), the angle α is the angle α₁₀₀. When the planeorientation of the silicon substrate 2 is (110), the angle α is theangle α₁₁₀. When the plane orientation of the silicon substrate 2 is(111), the angle α is the angle α₁₁₁.

Here, an example of the phase characteristics of an acoustic wave devicethat has a similar configuration to the first preferred embodiment andin which the angle α₁₁₁ is defined will be described. The designparameters of the acoustic wave device are as follows.

-   -   Silicon Substrate 2: Material monocrystal silicon, Plane        orientation (111), Euler angles (φ, θ, ψ) (−45°, −54.7°, 60°),        and Thickness 20 μm    -   Polysilicon Layer 3: Material polysilicon, and Thickness 1 μm    -   Silicon Oxide Layer 5: Material SiO₂, and Thickness 300 nm    -   Piezoelectric Layer 7: Material LiTaO₃, Cut Angle 40° Y, Euler        angles (φ, θ, ψ) (0°, 130°, 0°), and Thickness 400 nm    -   Layer Configuration of Interdigital Transducer Electrode 8:        Material Ti, AlCu, and Ti from the piezoelectric layer 7 side,        the content of Cu in AlCu is 1 wt %, and Thickness 12 nm, 100        nm, and 4 nm from the piezoelectric layer 7 side    -   Duty Ratio of Interdigital Transducer Electrode 8: 0.5    -   Wave Length λ of Interdigital Transducer Electrode 8: 2 μm    -   α₁₁₁: 60°

FIG. 16 is a graph that shows the phase characteristics of the acousticwave device of which α₁₁₁ is 60° in the first preferred embodiment.

As shown in FIG. 16 , it appears that a higher-order mode is reduced toless than −80[deg.] at not only frequencies around 1.5 times theresonant frequency but also frequencies around 2.2 times the resonantfrequency.

Incidentally, as shown in FIG. 1 , in the present preferred embodiment,the silicon oxide layer 5 is provided directly on the polysilicon layer3. The piezoelectric layer 7 is provided directly on the silicon oxidelayer 5. Of course, the silicon oxide layer 5 may be provided indirectlyon the polysilicon layer 3 with another layer interposed therebetween.Similarly, the piezoelectric layer 7 may be provided indirectly on thesilicon oxide layer 5 with another layer interposed therebetween.

FIG. 17 is an elevational cross-sectional view around a pair ofelectrode fingers of an acoustic wave device according to a secondpreferred embodiment.

The present preferred embodiment differs from the first preferredembodiment in that a silicon nitride layer 26 is provided between thesilicon oxide layer 5 and the piezoelectric layer 7. The presentpreferred embodiment further differs from the first preferred embodimentin that a protective film 29 is provided on the piezoelectric layer 7 soas to cover the interdigital transducer electrode 8. Other than theabove points, the acoustic wave device according to the presentpreferred embodiment has a similar configuration to the acoustic wavedevice 1 according to the first preferred embodiment.

Here, an example of the phase characteristics of an acoustic wave devicethat has a similar configuration to the present preferred embodiment andin which the angle α₁₁₁ is defined will be described. The designparameters of the acoustic wave device are similar to those of theacoustic wave device according to the first preferred embodiment inwhich the phase characteristics shown in FIG. 16 are measured, exceptthe following points.

-   -   Polysilicon Layer 3: Material polysilicon, Thickness 1.3 μm    -   Silicon Nitride Layer 26: Material SiN, and Thickness 50 nm    -   Layer Configuration of Interdigital Transducer Electrode 8:        Material Ti, AlCu, and Ti from the piezoelectric layer 7 side,        the content of Cu in AlCu is 1 wt %, and Thickness 10 nm, 100        nm, and 4 nm from the piezoelectric layer 7 side    -   Protective Film 29: Material SiO₂, and Thickness 30 nm α₁₁₁: 60°

FIG. 18 is a graph that shows the phase characteristics of the acousticwave device according to the second preferred embodiment.

As shown in FIG. 18 , it appears that, in the present preferredembodiment, higher-order modes are reduced in a wide band.

FIG. 19 is an elevational cross-sectional view around a pair ofelectrode fingers of an acoustic wave device according to a thirdpreferred embodiment.

The present preferred embodiment differs from the first preferredembodiment in that the silicon nitride layer 26 is provided between thepolysilicon layer 3 and the silicon oxide layer 5. The present preferredembodiment further differs from the first preferred embodiment in thatthe protective film 29 is provided on the piezoelectric layer 7 so as tocover the interdigital transducer electrode 8. Other than the abovepoints, the acoustic wave device according to the present preferredembodiment has a similar configuration to the acoustic wave device 1according to the first preferred embodiment.

Here, an example of the phase characteristics of an acoustic wave devicethat has a similar configuration to the present preferred embodiment andin which the angle α₁₁₁ is defined will be described. The designparameters of the acoustic wave device are similar to those of theacoustic wave device according to the first preferred embodiment inwhich the phase characteristics shown in FIG. 16 are measured, exceptthe following points.

-   -   Silicon Nitride Layer 26: Material SiN, and Thickness 50 nm    -   Layer Configuration of Interdigital Transducer Electrode 8:        Material Ti, AlCu, and Ti from the piezoelectric layer 7 side,        the content of Cu in AlCu is 1 wt %, and Thickness 10 nm, 100        nm, and 4 nm from the piezoelectric layer 7 side    -   Protective Film 29: Material SiO₂, and Thickness 30 nm α₁₁₁: 60°

FIG. 20 is a graph that shows the phase characteristics of the acousticwave device according to the third preferred embodiment.

As shown in FIG. 20 , it appears that, in the present preferredembodiment, higher-order modes are reduced in a wide band. In addition,in the present preferred embodiment, as show in FIG. 19 , the siliconnitride layer 26 is provided between the polysilicon layer 3 and thesilicon oxide layer 5. With this configuration, generation of electriccharge and electron transfer are reduced. Thus, degradation of IMDcharacteristics is reduced or prevented.

FIG. 21 is an elevational cross-sectional view around a pair ofelectrode fingers of an acoustic wave device according to a fourthpreferred embodiment.

The present preferred embodiment differs from the second preferredembodiment in that a titanium oxide layer 36 is provided between thesilicon oxide layer 5 and the piezoelectric layer 7. A multilayersubstrate 39 is a multilayer body of the silicon substrate 2, thepolysilicon layer 3, the silicon oxide layer 5, the titanium oxide layer36, and the piezoelectric layer 7. Other than the above points, theacoustic wave device according to the present preferred embodiment has asimilar configuration to the acoustic wave device according to thesecond preferred embodiment.

Here, an example of the phase characteristics of an acoustic wave devicethat has a similar configuration to the present preferred embodiment andin which the angle α₁₁₁ is defined will be described. The designparameters of the acoustic wave device are similar to those of theacoustic wave device according to the second preferred embodiment inwhich the phase characteristics shown in FIG. 18 are measured, exceptthe following points.

-   -   Polysilicon Layer 3: Material polysilicon, and Thickness 1 μm    -   Titanium Oxide Layer 36: Material TiO₂, and Thickness 30 nm    -   α₁₁₁: 30°

FIG. 22 is a graph that shows the phase characteristics of the acousticwave device according to the fourth preferred embodiment.

As shown in FIG. 22 , it appears that, in the present preferredembodiment, higher-order modes are reduced in a wide band. In addition,in the present preferred embodiment, as shown in FIG. 21 , the titaniumoxide layer 36 is provided between the silicon oxide layer 5 and thepiezoelectric layer 7. Since the titanium oxide layer 36 has a largedielectric constant, a fractional band width is narrowed.

Here, in each of the acoustic wave devices respectively having themultilayer substrates according to the first, second, and fourthpreferred embodiments, the phase of a higher-order mode was measuredwhile the parameters such as the angle α were changed. Thus, conditionsin which the phase of a higher-order mode was reduced to less than orequal to about −70[deg.] or less than or equal to about −80[deg.] wereobtained. In each of the acoustic wave devices, the protective film 29shown in FIG. 17 and the like is not provided. The conditions in whichhigher-order modes were reduced were obtained in each of a case wherethe plane orientation of the silicon substrate 2 was (100), a case wherethe plane orientation was (110), and a case where the plane orientationwas (111). Hereinafter, the details will be described.

In the configuration of the first preferred embodiment shown in FIG. 1 ,the design parameters and the variable ranges of the design parameterswere set as follows. The plane orientation of the silicon substrate 2was set to (100).

-   -   Silicon Substrate 2: Material monocrystal silicon, Plane        orientation (100), and Thickness 20 μm    -   Polysilicon Layer 3: Material polysilicon, and Thickness changed        in increments of 0.1 μm in the range greater than or equal to        0.1 μm and less than or equal to 1.5 μm.    -   Silicon Oxide Layer 5: Material SiO₂, and Thickness changed in        increments of 0.05 μm in the range greater than or equal to 0.2        μm and less than or equal to 0.4 μm    -   Piezoelectric Layer 7: Material LiTaO₃, Cut Angle 40° Y, Euler        angles (φ, θ, ψ) (0°, 130°, 0°), and Thickness changed in        increments of 0.1 μm in the range greater than or equal to 0.3        μm and less than or equal to 0.4 μm    -   Layer Configuration of Interdigital Transducer Electrode 8:        Material Ti, AlCu, and Ti from the piezoelectric layer 7 side,        the content of Cu in AlCu is 1 wt %, and Thickness 12 nm, 100        nm, and 4 nm from the piezoelectric layer 7 side    -   Duty Ratio of Interdigital Transducer Electrode 8: 0.5    -   Wave Length λ of Interdigital Transducer Electrode 8: 2 μm    -   α₁₀₀: changed in increments of 5° in the range greater than or        equal to 0° and less than or equal to 45°

There is a four-fold symmetry in the (100) plane of a silicon substrate,and an equivalent crystal structure is obtained by 90° rotation. Thus,when the plane orientation of the silicon substrate 2 is (100), theangle α₁₀₀ is set to α₁₀₀=α₁₀₀+90×n. n is an integer (0, ±1, ±2, . . .).

The phase of a higher-order mode was measured while the parameters werechanged as described above. Thus, the equation 1 that was a relationalexpression between the parameters and the phase of a higher-order modewas derived. The angle α is assumed as Si_psi[deg.], the thickness ofthe piezoelectric layer 7 is assumed as t_LT[λ], the thickness of thesilicon oxide layer 5 is assumed as t_SiO₂[λ], the thickness of thepolysilicon layer 3 is assumed as t_Si₂[λ], and the phase of ahigher-order mode is assumed as y[deg.]. In the equation 1, Si_psi[deg.]is the angle α₁₀₀. In the equations in the specification, unit [deg.]represents the same meaning as unit [°].

y[deg.]=(−72.1492542241195)+0.627588217157224×(Si_psi[deg]−21.7083333333333)+(−1.93347870945237)×(t_Si₂[λ]−0.4525)+72.3846086764674×(t_LT[λ]−0.160833333333333)+(−67.3219584197057)×(t_SiO₂[λ]−0.16625)+0.0000655654050315201×((Si_psi[deg.]−21.7083333333333)×(Si_psi[deg.]−21.7083333333333)−25.2065972222222)+(−2.34857364418332)×((Si_psi[deg.]−21.7083333333333)×((t_Si₂[λ]−0.4525))+37.0048979126418×((t_Si ₂[λ]−0.4525)×((t_Si₂[λ]−0.4525)−0.0360354166666667)+7.0771357128953×((Si_psi[deg.]−21.7083333333333)×((t_LT[λ]−0.160833333333333))+(−10.057857939681)×((t_Si₂[λ]−0.4525)×(t_LT[λ]−0.160833333333333))+1.50716777611893×((Si_psi[deg.]−21.7083333333333)×(t_SiO₂[λ]−0.16625))+426.86632497558×(((t_Si ₂[λ]−0.4525)×((t_SiO₂[λ])−0.16625))+925.280868396996×((t_LT[λ]−0.160833333333333)×(t_SiO₂[λ]−0.16625))+988.798729044457×((t_SiO ₂[λ]−0.16625)×(t_SiO₂[λ]−0.16625)−0.000871354166666668)  Equation 1

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], and t_Si₂[λ] each are preferably avalue within a range with which y in the equation 1 is less than orequal to about −70. With this configuration, the phase of a higher-ordermode is further reliably set to less than or equal to about −70 [deg.].Therefore, higher-order modes are further reliably and effectivelyreduced.

In the configuration of the first preferred embodiment, the planeorientation of the silicon substrate 2 was set to (110), and the phaseof a higher-order mode was measured while the parameters were changed.The design parameters and the variable ranges of the design parameterswere similar to those when the equation 1 was derived except the angleα.

-   -   α₁₁₀: changed in increments of 10° in the range greater than or        equal to 0° and less than or equal to 90°

There is a two-fold symmetry in the (110) plane of a silicon substrate,and an equivalent crystal structure is obtained by 180° rotation. Thus,when the plane orientation of the silicon substrate 2 is (110), theangle α₁₁₀ is set to α₁₁₀=α₁₁₀+180×n. n is an integer (0, ±1, 2, . . .).

The phase of a higher-order mode was measured while the parameters werechanged as described above. Thus, the equation 2 that was a relationalexpression between the parameters and the phase of a higher-order modewas derived. In the equation 2, Si_psi [deg.] is the angle α₁₁₀.

y[deg.]=(78.1876454049157)+(−0.182894322081067)×(Si_psi[deg.]−28.1088082901554)+6.18390256271178×(t_Si₂[λ]−0.39961139896373)+116.669335737855×(t_LT[λ]−0.169948186528498)+10.3573467893808×(t_SiO₂[λ]−0.144041450777202)+0.0110735958981267×((Si_psi[deg.]−28.1088082901554)×(Si_psi[deg.]−28.1088082901554)−189.946709978791)+(−0.246858144090431)×((Si_psi[deg.]−28.1088082901554)×(t_Si₂[λ]−0.39961139896373)+22.031016276383×((t_Si₂[λ]−0.39961139896373)×(t_Si₂[λ]−0.39961139896373)−0.0484389681602191)+(−X.0545756011518778)×((Si_psi[deg.]−28.1088082901554)×(t_LT[λ]−0.169948186528498))+(−32.427969747408)×((t_Si₂[λ]−0.39961139896373)×((t_LT[λ]−0.169948186528498))+(−2.62164982026802)×((Si_psi[deg.]−28.1088082901554)×(t_SiO₂[λ]−0.144041450777202))+(−112.759047075747)×((t_Si₂[λ]−0.39961139896373)×((t_SiO₂[λ]−0.144041450777202))+(−604.832727678973)×((t_LT[λ]−0.169948186528498)×((t_SiO₂[λ]−0.144041450777202))+326.415587634024×((t_SiO₂[λ]−0.144041450777202)×((t_SiO₂[λ]−0.144041450777202)−0.00120154232328385)  Equation 2

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], and t_Si₂[λ] each are preferably avalue within a range with which y in the equation 2 is less than orequal to about −70. With this configuration, the phase of a higher-ordermode is further reliably set to less than or equal to about −70 [deg.].Therefore, higher-order modes are further reliably and effectivelyreduced.

In the configuration of the first preferred embodiment, the planeorientation of the silicon substrate 2 was set to (111), and the phaseof a higher-order mode was measured while the parameters were changed.The design parameters and the variable ranges of the design parameterswere similar to those when the equation 1 was derived except the angleα.

-   -   α₁₁₁: changed in increments of 5° in the range greater than or        equal to 0° and less than or equal to 60°

There is a three-fold symmetry in the (111) plane of a siliconsubstrate, and an equivalent crystal structure is obtained by 120°rotation. Thus, when the plane orientation of the silicon substrate 2 is(111), the angle α₁₁₁ is set to α₁₁₁=α₁₁₁+120×n. n is an integer (0, ±1,2, . . . ).

The phase of a higher-order mode was measured while the parameters werechanged as described above. Thus, the equation 3 that was a relationalexpression between the parameters and the phase of a higher-order modewas derived. In the equation 3, Si_psi [deg.] is the angle α₁₁₁.

y[deg.]=(77.9109394183719)+(−0.0492368384201428)×(Si_psi[deg.]−45.2068126520681)+0.525124426223863×(t_Si₂[λ]−0.426216545012165)+117.400884406373×(t_LT[λ]−0.174330900243311)+(−2.62484877324049)×(t_SiO₂[λ]−0.15139902676399)+0.00307563131201403×((Si_psi[deg.]−45.2068126520681)×(Si_psi[deg.]−45.2068126520681)−182.925598356629)+(−0.0261801752592506)×((Si_psi[deg.]−45.2068126520681)×(t_Si₂[λ]−0.426216545012165))+23.8987529211434×((t_Si₂[λ]−0.426216545012165)×(t_Si₂[λ]−0.426216545012165)−0.0481296027432942)+1.52616542281399×((Si_psi[deg.]−45.2068126520681))×(t_LT[λ]−0.174330900243311))+(−129.002027283367)×((t_Si₂[λ]−0.426216545012165)×((t_LT[λ]−0.174330900243311))+(−1.22761778451819)×((Si_psi[deg.]−45.2068126520681)×((t_SiO₂[λ]−0.15139902676399))+(−42.6041784800926)×((t_Si₂[λ]−0.426216545012165)×(t_SiO₂[λ]−0.15139902676399))+(−468.84116493048)×((t_LT[λ]−0.174330900243311)×(t_SiO₂[λ]−0.15139902676399))+(−8.20635607220859)×((t_SiO₂[λ]−0.15139902676399)×(t_SiO₂[λ]−0.15139902676399)−0.0012830183932134)  Equation 3

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], and t_Si₂[λ] each are preferably avalue within a range with which y in the equation 3 is less than orequal to about −70. With this configuration, the phase of a higher-ordermode is further reliably set to less than or equal to about −70 [deg.].Therefore, higher-order modes are further reliably and effectivelyreduced.

In addition, conditions in which the phase of a higher-order mode wasreduced to less than or equal to about −80[deg.] were obtained in eachof a case where the plane orientation of the silicon substrate 2 was(100), a case where the plane orientation of the silicon substrate 2 was(110), and a case where the plane orientation of the silicon substrate 2was (111). In these cases, the relational expression between theparameters and a higher-order mode differs from the equation 1, theequation 2, or the equation 3. More specifically, to obtain theconditions, an equation 4, an equation 5, and an equation 6 were derivedwhile the parameters were changed within the range in which the phase ofa higher-order mode was greater than or equal to −90[deg.] and less thanor equal to about −70[deg.].

When the plane orientation of the silicon substrate 2 is (100), theequation 4 was derived as described above.

y[deg.]=(−75.3156232479379)+0.63547968892276×(Si_psi[deg]−20.9090909090909)+(−2.02838142816204)×(t_Si₂[λ]−0.439772727272727)+90.1874317877843×(t_LT[λ]−0.151136363636364)+(−71.2997621594781)×(t_SiO₂[λ]−0.171590909090909)+0.108397383766316×((Si_psi[deg.]−20.9090909090909)×(Si_psi[deg.]−20.9090909090909)−13.9462809917355)+(−3.76982864951476)×((Si_psi[deg.]−20.9090909090909)×(t_Si₂[λ]−0.439772727272727))+37.3378798744213×((t_Si₂[λ]−0.439772727272727)×(t_Si₂[λ]−0.439772727272727)−0.0358613119834711)+(−23.7942425679855)×((Si_psi[deg.]−20.9090909090909)×(t_SiO₂[λ]−0.171590909090909))+462.018905986831×((t_Si₂[λ])−0.439772727272727)×(t_SiO₂[λ]−0.171590909090909))+1223.13016730739×(((t_SiO₂[λ]−0.171590909090909)×(t_SiO₂[λ]−0.171590909090909)−0.000641787190082645)  Equation 4

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], and t_Si₂[λ] each are preferably avalue within a range with which y in the equation 4 is less than orequal to about −80. With this configuration, the phase of a higher-ordermode is further reliably set to less than or equal to about −80[deg.].Therefore, higher-order modes are further reliably and further reduced.

When the plane orientation of the silicon substrate 2 is (110), theequation 5 was derived as described above.

y[deg.]=(81.4138269086073)+(−0.100532115186538)×(Si_psi[deg.]−29.1379310344828)+0.845708574223377×(t_Si₂[λ]−0.385689655172414)+87.6682874459356×(t_LT[λ]−0.166724137931034)+(−0.137780433371857)×(t_SiO₂[λ]−0.145)+0.00337749443465239×((Si_psi[deg.]−29.1379310344828)×(Si_psi[deg.]−29.1379310344828)−127.877526753864)+(−0.116548121456389)×((Si_psi[deg.]−29.1379310344828)×(t_Si₂[λ]−0.385689655172414))+11.8893452691356×((t_Si₂[λ]−0.385689655172414)×(t_Si₂[λ]−0.385689655172414)−0.0448900416171225)+0.333200244545922×((Si_psi[deg.]−29.1379310344828)×(t_LT[λ]−0.166724137931034))+55.2630600466406×((t_Si₂[λ]−0.385689655172414)×(t_LT[λ]−0.166724137931034))+(−0.296582437395607)×((Si_psi[deg.]−29.1379310344828)×(t_SiO₂[λ]−0.145))+(−67.4578937630203)×((t_Si ₂[λ]−0.385689655172414)×(t_SiO₂[λ]−0.145))+(−376.292315976729)×((t_LT[λ])−0.166724137931034)×(t_SiO₂[λ]−0.145))+48.6290874437329×((t_SiO ₂[λ]−0.145)×((t_SiO₂[λ]−0.145)−0.00120775862068966)  Equation 5

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], and t_Si₂[λ] each are preferably avalue within a range with which y in the equation 5 is less than orequal to about −80. With this configuration, the phase of a higher-ordermode is further reliably set to less than or equal to about −80[deg.].Therefore, higher-order modes are further reliably and further reduced.

When the plane orientation of the silicon substrate 2 is (111), theequation 6 was derived as described above.

y[deg.]=(−79.8924944284088)+0.033261334588906×(Si_psi[deg.]−39.3173431734317)+3.93783296791666×(t_Si₂[λ]−0.416974169741698)+80.6680077909648×(t_LT[λ]−0.17140221402214)+13.2276438709535×(t_SiO₂[λ]−0.148431734317343)+(−0.00907764275073328)×((Si_psi[deg.]−39.3173431734317)×(Si_psi[deg.]−39.3173431734317)−21.2129464468077)+0.000540095694459618×((Si_psi[deg.]−39.3173431734317)×(t_Si₂[λ]−0.416974169741698))+5.79698263968963×((t_Si₂[λ]−0.416974169741698)×(t_Si₂[λ]−0.416974169741698)−0.0400439808826132)+(−0.136650035849863)×((Si_psi[deg.]−39.3173431734317)×(t_LT[λ]−0.17140221402214))+(−20.3328823416631)×((t_Si₂[λ]−0.416974169741698)×(t_LT[λ]−0.17140221402214))+(−2.22480760136672)×((Si_psi[deg.]−39.3173431734317)×(t_SiO₂[λ]−0.148431734317343))+(−13.0975601885972)×((t_Si₂[λ]−0.416974169741698)×(t_SiO₂[λ]−0.148431734317343))+(−511.743077543129)×((t_LT[λ]−0.17140221402214)×(t_SiO₂[λ]−0.148431734317343))+137.213612130809×((t_SiO₂[λ]−0.148431734317343)×(t_SiO₂[λ]−0.148431734317343)−0.00135593537669694)  Equation 6

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], and t_Si₂[λ] each are preferably avalue within a range with which y in the equation 6 is less than orequal to about −80. With this configuration, the phase of a higher-ordermode is further reliably set to less than or equal to about −80[deg.].Therefore, higher-order modes are further reliably and further reduced.

Subsequently, in the configuration having a multilayer substrate similarto that of the second preferred embodiment shown in FIG. 17 , the designparameters and the variable ranges of the design parameters were set asfollows. The plane orientation of the silicon substrate 2 was set to(100).

-   -   Silicon Substrate 2: Material monocrystal silicon, Plane        orientation (100), and Thickness 20 μm    -   Polysilicon Layer 3: Material polysilicon, and Thickness changed        in increments of 0.1 μm in the range greater than or equal to        0.1 μm and less than or equal to 1.5 μm.    -   Silicon Oxide Layer 5: Material SiO₂, and Thickness changed in        increments of 0.05 μm in the range greater than or equal to 0.2        μm and less than or equal to 0.4 μm    -   Silicon Nitride Layer 26: Material SiN, Thickness changed in        increments of 0.02 μm in the range greater than or equal to 0.01        μm and less than or equal to 0.15 μm.    -   Piezoelectric Layer 7: Material LiTaO₃, Cut Angle 40° Y, Euler        angles (φ, θ, ψ) (0°, 130°, 0°), and Thickness changed in        increments of 0.1 μm in the range greater than or equal to 0.3        μm and less than or equal to 0.4 μm    -   Layer Configuration of Interdigital Transducer Electrode 8:        Material Ti, AlCu, and Ti from the piezoelectric layer 7 side,        the content of Cu in AlCu is 1 wt %, and Thickness 12 nm, 100        nm, and 4 nm from the piezoelectric layer 7 side    -   Duty Ratio of Interdigital Transducer Electrode 8: 0.5    -   Wave Length λ of Interdigital Transducer Electrode 8: 2 μm    -   α₁₀₀: changed in increments of 5° in the range greater than or        equal to 0° and less than or equal to 450

The phase of a higher-order mode was measured while the parameters werechanged as described above. Thus, an equation 7 that was a relationalexpression between the parameters and the phase of a higher-order modewas derived. The thickness of the silicon nitride layer 26 is set tot_SiN[λ]. In the equation 7, Si_psi[deg.] is the angle α₁₀₀.

y[deg.]=(−67.7782730918073)+0.0667732718475358×(Si_psi[deg.]−25.6259314456036)+(−6.71256568714434)×(t_Si₂[λ]−0.426192250372578)+177.355083873051×(t_LT[λ]−0.16602086438151)+(−64.7093491078986)×(t_SiN[λ]−0.0465201192250378)+1.0890884781807×(t_SiO₂[λ]−0.155793591654245)+0.000179985859065592×(Si_psi[deg.]−25.6259314456036)×(Si_psi[deg.]−25.6259314456036)−130.38317256758)+(−0.329348427439478)×((Si_psi[deg.]−25.6259314456036)×(t_Si₂[λ]−0.426192250372578))+(−33.1084698932093)×((t_Si₂[λ]−0.426192250372578)×(t_Si₂[λ]−0.426192250372578)−0.0504801359160987)+1.52146775761601×((Si_psi[deg.]−25.6259314456036)×(t_LT[λ]−0.16602086438151))+14.59741625744683×((t_Si₂[λ]−0.426192250372578)×(t_LT[λ]−0.16602086438151))+0×((t_LT[λ]−0.16602086438151)×(t_LT[λ]−0.16602086438151)−0.000544375123544922)+(−4.94058423048505)×((Si_psi[deg.]−25.6259314456036)×((t_SiN[λ]−0.0465201192250378))+138.799085167873×((t_Si₂[λ]−0.426192250372578)×(t_SiN[λ]−0.0465201192250378))+1746.7447498235×((t_LT[λ]−0.16602086438151)×(t_SiN[λ]−0.0465201192250378))+2167.04168685901×((t_SiN[λ]−0.0465201192250378)×(t_SiN[λ]−0.0465201192250378)−0.000465274930537198)+(−0.931372972560935)×((Si_psi[deg.]−25.6259314456036)×(t_SiO₂[λ]−0.155793591654245))+(−79.4377446578721)×((t_Si₂[λ]−0.426192250372578)×(t_SiO₂[λ]−0.155793591654245))+(−86.9697272546991)×((t_LT[λ]−0.16602086438151)×(t_SiO₂[λ]−0.155793591654245))+1966.46522796354×((t_SiN[λ]−0.0465201192250378)×(t_SiO₂[λ]−0.155793591654245))+169.040605778099×((t_SiO₂[λ]−0.155793591654245)×(t_SiO₂[λ]−0.155793591654245)-0.00164210493657841)  Equation 7

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], t_SiN[λ], and t_Si₂[λ] each arepreferably a value within a range with which y in the equation 7 is lessthan or equal to about −70. With this configuration, the phase of ahigher-order mode is further reliably set to less than or equal to about−70[deg.]. Therefore, higher-order modes are further reliably andeffectively reduced.

In the configuration having a multilayer substrate similar to that ofthe second preferred embodiment, the plane orientation of the siliconsubstrate 2 was set to (110), and the phase of a higher-order mode wasmeasured while the parameters were changed. The design parameters andthe variable ranges of the design parameters were similar to those whenthe equation 7 was derived except the angle α.

α₁₁₀: changed in increments of 10° in the range greater than or equal to0° and less than or equal to 90°

The phase of a higher-order mode was measured while the parameters werechanged as described above. Thus, an equation 8 that was a relationalexpression between the parameters and the phase of a higher-order modewas derived. In the equation 8, Si_psi [deg.] is the angle α₁₁₀.

y[deg.]=(−75.0174122935603)+(−0.00810936153116664)×(Si_psi[deg.]−42.0340722495895)+1.98135617767495×(t_Si₂[λ]−0.385026683087027)+143.173790020328×(t_LT[λ]−0.17306034482757)+16.4148627328736×(t_SiN[λ]−0.04207922824302)+50.4122771861205×(t_SiO₂[λ])−0.144909688013139)+0.00619821963137332×((Si_psi[deg.]−42.0340722495895)×(Si_psi[deg.]−42.0340722495895)−514.232829229589)+0.020323078287526×((Si_psi[deg.]−42.0340722495895)×(t_Si₂[λ]−0.385026683087027))+1.15443318031007×((t_Si₂[λ]−0.385026683087027)×(t_Si₂[λ]−0.385026683087027)−0.0477966331139576)+0.472662465737381×((Si_psi[deg.]−42.0340722495895)×(t_LT[λ]−0.17306034482757))+(−105.2996012677)×((t_Si₂[λ]−0.385026683087027)×(t_LT[λ]−0.17306034482757))+(−1.29517116632701)×((Si_psi[deg.]−42.0340722495895)×(t_SiN[λ]−0.04207922824302))+(−26.1801037669841)×((t_Si₂[λ]−0.385026683087027)×(t_SiN[λ])−0.04207922824302))+168.1334353773×((t_LT[λ]−0.17306034482757)×(t_SiN[λ]−0.04207922824302))+2120.76431830662×((t_SiN[λ]−0.04207922824302)×(t_SiN[λ]−0.04207922824302)−0.000508197335364991)+(−0.687562974959064)×((Si_psi[deg.]−42.0340722495895)×(t_SiO₂[λ]−0.144909688013139))+15.3482271106745×((t_Si₂[λ]−0.385026683087027)×(t_SiO₂[λ]−0.144909688013139))+(−358.720795782422)×((t_LT[λ]−0.17306034482757)×(t_SiO₂[λ]−0.144909688013139))+1062.30534015379×((t_SiN[λ]−0.04207922824302)×(t_SiO₂[λ]−0.144909688013139))+248.937429294479×((t_SiO₂[λ]−0.144909688013139)×(t_SiO₂[λ]−0.144909688013139)−0.00162330875671721)  Equation 8

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], t_SiN[λ], and t_Si₂[λ] each arepreferably a value within a range with which y in the equation 8 is lessthan or equal to about −70. With this configuration, the phase of ahigher-order mode is further reliably set to less than or equal to about−70[deg.]. Therefore, higher-order modes are further reliably andeffectively reduced.

In the configuration having a multilayer substrate similar to that ofthe second preferred embodiment, the plane orientation of the siliconsubstrate 2 was set to (111), and the phase of a higher-order mode wasmeasured while the parameters were changed. The design parameters andthe variable ranges of the design parameters were similar to those whenthe equation 7 was derived except the angle α. α₁₁₁: changed inincrements of 5° in the range greater than or equal to 0° and less thanor equal to 60°

The phase of a higher-order mode was measured while the parameters werechanged as described above. Thus, an equation 9 that was a relationalexpression between the parameters and the phase of a higher-order modewas derived. In the equation 9, Si_psi [deg.] is the angle α₁₁₁.

y[deg.]=(−77.5405307874512)+0.00496521862619995×(Si_psi[deg.]−44.3479880774963)+(−3.07514699616305)×(t_Si₂[λ]−0.395628415300543)+115.725430166886×(t_LT[λ]−0.173919523099848)+75.6109484741613×(t_SiN[λ]−0.0387729756582212)+29.9143205043822×(t_SiO₂[λ]−0.145404868355688)+0.00452378218877289×((Si_psi[deg.]−44.3479880774963)×(Si_psi[deg.]−44.3479880774963)−147.519490487682)+(−0.127045459018856)×((Si_psi[deg.]−44.3479880774963)×(t_Si₂[λ]−0.395628415300543))+10.135015813019×((t_Si₂[λ]−0.395628415300543)×(t_Si₂[λ]−0.395628415300543)−0.0544331992323139)+0.267609205446981×((Si_psi[deg.]−44.3479880774963)×(t_LT[λ]−0.173919523099848))+(−151.966315117959)×((t_Si₂[λ]−0.395628415300543)×(t_LT[λ]−0.173919523099848))+1.1818941610908×((Si_psi[deg.]−44.3479880774963)×(t_SiN[λ]−0.0387729756582212))+(−19.0228093275549)×((t_Si₂[λ]−0.395628415300543)×(t_SiN[λ]−0.0387729756582212))+25.2693219567039×((t_LT[λ]−0.173919523099848)×(t_SiN[λ])−0.0387729756582212))+1545.52112794945×((t_SiN[λ]−0.0387729756582212)×(t_SiN[λ]−0.0387729756582212)−0.000519520243356094)+(−0.39161225199813)×((Si_psi[deg.]−44.3479880774963)×(t_SiO₂[λ]−0.145404868355688))+22.0391330835907×((t_Si₂[λ]−0.395628415300543)×(t_SiO₂[λ]−0.145404868355688))+(−297.764935637906)×((t_LT[λ]−0.173919523099848)×(t_SiO₂[λ]−0.145404868355688))+982.324171494675×((t_SiN[λ]−0.0387729756582212)×(t_SiO₂[λ]−0.145404868355688))+420.570041600812×((t_SiO₂[λ]−0.145404868355688)×(t_SiO₂[λ]−0.145404868355688)−0.00124068307615005)  Equation 9

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], t_SiN[λ], and t_Si₂[λ] each arepreferably a value within a range with which y in the equation 9 is lessthan or equal to about −70. With this configuration, the phase of ahigher-order mode is further reliably set to less than or equal to about−70[deg.]. Therefore, higher-order modes are further reliably andeffectively reduced.

In addition, conditions in which the phase of a higher-order mode wasreduced to less than or equal to about −80[deg.] were obtained in eachof a case where the plane orientation of the silicon substrate 2 was(100), a case where the plane orientation of the silicon substrate 2 was(110), and a case where the plane orientation of the silicon substrate 2was (111). More specifically, to obtain the conditions, an equation 10,an equation 11, and an equation 12 were derived while the parameterswere changed within the range in which the phase of a higher-order modewas greater than or equal to −90[deg.] and less than or equal to about−70[deg.].

When the plane orientation of the silicon substrate 2 is (100), theequation 10 was derived as described above.

y[deg.]=(−78.3557914112162)+(−0.00785147182473267)×(Si_psi[deg.]−24.9802110817942)+(−1.32878861667394)×(t_Si₂[λ]−0.429221635883905)+(−41.7937386863014)×(t_LT[λ]−0.150923482849606)+35.6722090195008×(t_SiN[λ]−0.0500263852242746)+18.7743164986736×(t_SiO₂[λ]−0.145646437994723)+(−0.000765722206063909)×((Si_psi[deg.]−24.9802110817942)×(Si_psi[deg.]−24.9802110817942)−153.396706024045)+(−0.0463379291760545)×((Si_psi[deg.]−24.9802110817942)×(t_Si₂[λ]−0.429221635883905))+(−17.7293821535291)×((t_Si₂[λ]−0.429221635883905)×(t_Si₂[λ]−0.429221635883905)−0.0593208981070862)+(−1.3441873888418)×((Si_psi[deg.]−24.9802110817942)×(t_LT[λ]−0.150923482849606))+(−417.636233521175)×((t_Si₂[λ]−0.429221635883905)×(t_LT[λ]−0.150923482849606))+(−0.487351707638102)×((Si_psi[deg.]−24.9802110817942)×(t_SiN[λ]−0.0500263852242746))+(−25.3025544220714)×((t_Si₂[λ]−0.429221635883905)×(t_SiN[λ]−0.0500263852242746))+1666.3381560311×((t_LT[λ]−0.150923482849606)×(t_SiN[λ]−0.0500263852242746))+233.559062145034×((t_SiN[λ]−0.0500263852242746)×(t_SiN[λ]−0.0500263852242746)−0.000389115398806747)+(−0.148028298904273)×((Si_psi[deg.]−24.9802110817942)×(t_SiO₂[λ]−0.145646437994723))+(−63.9722673973965)×((t_Si×[λ]−0.429221635883905)×(t_SiO₂[λ]−0.145646437994723))+1197.10044921435×((t_LT[λ]−0.150923482849606)×(t_SiO₂[AX]−0.145646437994723))+450.45656510444×((t_SiN[λ]−0.0500263852242746)×(t_SiO₂ [AX])−0.145646437994723))+(−37.7857111587959)×((t_SiO₂[λ]−0.145646437994723)×(t_SiO ₂[AX]−0.145646437994723)−0.0017158749939084)  Equation 10

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], t_SiN[λ], and t_Si₂[λ] each arepreferably a value within a range with which y in the equation 10 isless than or equal to about −80. With this configuration, the phase of ahigher-order mode is further reliably set to less than or equal to about−80[deg.]. Therefore, higher-order modes are further reliably andfurther reduced.

When the plane orientation of the silicon substrate 2 is (110), theequation 11 was derived as described above.

y[deg.]=(−79.9409825800918)+0.00367175250563163×(Si_psi[deg.]−42.1225309675259)+(−1.19942177285592)×(t_Si₂[λ]−0.381570137261466)+91.8359644721651×(t_LT[λ]−0.164596585202533)+58.8431912005245×(t_SiN[λ]−0.0395698024774026)+16.9153289429696×(t_SiO₂[λ]−0.13875125544024)+0.00130491910714855×((Si_psi[deg.]−42.1225309675259)×(Si_psi[deg.]−42.1225309675259)−385.786124427809)+0.0745672315210127×((Si_psi[deg]−42.1225309675259)×(t_Si₂[λ]−0.381570137261466))+2.6699307571413×((t_Si₂[λ]−0.381570137261466)×(t_Si₂[λ]−0.381570137261466)−0.0456605075514713)+(−0.377889849052574)×((Si_psi[deg.]−42.1225309675259)×(t_LT[λ]−0.164596585202533))+(−43.4148735553507)×((t_Si₂[λ]−0.381570137261466)×(t_LT[λ]−0.164596585202533))+(−0.378387168121428)×((Si_psi[deg.]−42.1225309675259)×(t_SiN[λ]−0.0395698024774026))+(−20.545088460627)×((t_Si₂[λ]−0.381570137261466)×(t_SiN[λ]−0.0395698024774026))+232.919108783203×((t_LT[λ]−0.164596585202533)×(t_SiN[λ]−0.0395698024774026))+840.791113736585×((t_SiN[λ]−0.0395698024774026)×(t_SiN[λ]−0.0395698024774026)−0.000464855104179262)+0.190837727117146×((Si_psi[deg.]−42.1225309675259)×(t_SiO₂[λ]−0.13875125544024))+0.695837098714372×((t_Si₂[λ]−0.381570137261466)×(t_SiO₂[λ]−0.13875125544024))+(−184.621593720628)×((t_LT[λ]−0.164596585202533)×(t_SiO₂[λ]−0.13875125544024))+607.033426600094×((t_SiN[λ]−0.0395698024774026)×(t_SiO₂[λ]−0.13875125544024))+142.721242732228×((t_SiO₂[λ]−0.13875125544024)×(t_SiO₂[λ]−0.13875125544024)−0.00152562510304392)  Equation 11

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], t_SiN[λ], and t_Si₂[λ] each arepreferably a value within a range with which y in the equation 11 isless than or equal to about −80. With this configuration, the phase of ahigher-order mode is further reliably set to less than or equal to about−80[deg.]. Therefore, higher-order modes are further reliably andfurther reduced.

When the plane orientation of the silicon substrate 2 is (111), theequation 12 was derived as described above.

y[deg.]=(−79.8683540124538)+0.0118371753456289×(Si_psi[deg.]−44.4595052524568)+(−1.99138796522555)×(t_Si₂[λ]−0.413673331074209)+88.0775643151379×(t_LT[λ]−0.167705862419511)+46.4734172707698×(t_SiN[λ]−0.0351321585903086)+14.4134894109961×(t_SiO₂[λ]−0.142222975262623)+0.00167085752221365×((Si_psi[deg.]−44.4595052524568)×(Si_psi[deg.]−44.4595052524568)−128.282924729805)+(−0.0463012101323173)×((Si_psi[deg.]−44.4595052524568)×(t_Si₂[λ]−0.413673331074209))+4.58192618035487×((t_Si₂[λ]−0.413673331074209)×(t_Si₂[λ]−0.413673331074209)−0.05167257915661)+0.524887931323933×((Si_psi[deg]−44.4595052524568)×(t_LT[λ]−0.167705862419511))+(−71.7492658390069)×((t_Si₂[λ]−0.413673331074209)×(t_LT[λ]−0.167705862419511))+0.73863390529294×((Si_psi[deg.]−44.4595052524568)×(t_SiN[λ]−0.0351321585903086))+(−42.8957552454222)×((t_Si₂[λ]−0.413673331074209)×(t_SiN[λ]−0.0351321585903086))+(−411.839865840595)×((t_LT[λ]−0.167705862419511)×(t_SiN[λ]−0.0351321585903086))+982.235412331017×((t_SiN[λ]−0.0351321585903086)×(t_SiN[λ]−0.0351321585903086)−0.000477142818756284)+(−0.236509133242243)×((Si_psi[deg.]−44.4595052524568)×(t_SiO₂[λ]−0.142222975262623))+17.2370398551984×((t_Si₂[λ]−0.413673331074209)×(t_SiO₂[λ]−0.142222975262623))+(−469.933137492789)×((t_LT[λ]−0.167705862419511)×(t_SiO₂[λ]−0.142222975262623))+541.748349798792×((t_SiN[λ]−0.0351321585903086)×(t_SiO₂[λ]−0.142222975262623))+226.311489477246×((t_SiO₂[λ]−0.142222975262623)×(t_SiO₂[λ]−0.142222975262623)−0.00116579711361478)  Equation 12

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], t_SiN[λ], and t_Si₂[λ] each arepreferably a value within a range with which y in the equation 12 isless than or equal to about −80. With this configuration, the phase of ahigher-order mode is further reliably set to less than or equal to about−80[deg.]. Therefore, higher-order modes are further reliably andfurther reduced.

Subsequently, in the configuration having the multilayer substrate 39similar to that of the fourth preferred embodiment shown in FIG. 21 ,the design parameters and the variable ranges of the design parameterswere set as follows. The plane orientation of the silicon substrate 2was set to (100).

-   -   Silicon Substrate 2: Material monocrystal silicon, Plane        orientation (100), and Thickness 20 μm    -   Polysilicon Layer 3: Material polysilicon, and Thickness changed        in increments of 0.1 μm in the range greater than or equal to        0.1 μm and less than or equal to 1.5 μm.    -   Silicon Oxide Layer 5: Material SiO₂, and Thickness changed in        increments of 0.05 μm in the range greater than or equal to 0.2        μm and less than or equal to 0.4 μm    -   Titanium Oxide Layer 36: Material TiO₂, Thickness changed in        increments of 0.02 μm in the range greater than or equal to 0.01        μm and less than or equal to 0.15 μm.    -   Piezoelectric Layer 7: Material LiTaO₃, Cut Angle 40° Y, Euler        angles (φ, θ, ψ) (0°, 130°, 0°), and Thickness changed in        increments of 0.1 μm in the range greater than or equal to 0.3        μm and less than or equal to 0.4 μm    -   Layer Configuration of Interdigital Transducer Electrode 8:        Material Ti, AlCu, and Ti from the piezoelectric layer 7 side,        the content of Cu in AlCu is 1 wt %, and Thickness 12 nm, 100        nm, and 4 nm from the piezoelectric layer 7 side    -   Duty Ratio of Interdigital Transducer Electrode 8: 0.5    -   Wave Length λ of Interdigital Transducer Electrode 8: 2 μm    -   α₁₀₀: changed in increments of 5° in the range greater than or        equal to 0° and less than or equal to 45°

The phase of a higher-order mode was measured while the parameters werechanged as described above. Thus, an equation 13 that was a relationalexpression between the parameters and the phase of a higher-order modewas derived. The thickness of the titanium oxide layer 36 is set tot_TiO₂[λ]. In the equation 13, Si_psi[deg.] is the angle α₁₀₀.

y[deg.]=(−47.9946211404703)+1.21901050350713×(Si_psi[deg.]−19.0566037735849)+4.12041154986452×(t_Si₂[λ]−0.408490566037736)+228.432202102143×(t_TiO₂[λ]−0.0288364779874214)+(−33.1253993677708)×(t_SiO₂[λ]−0.160062893081761)+0.140472008263765×((Si_psi[deg,]−19.0566037735849)×(Si_psi[deg.]−19.0566037735849)−38.1037142518096)+(−5.44594625372052)×((Si_psi[deg.]−19.0566037735849)×(t_Si₂[λ]−0.408490566037736))+114.747133042737×((t_Si₂[λ]−0.408490566037736)×(t_Si₂[λ]−0.408490566037736)−0.0406983505399312)+10.4171695197979×((Si_psi[deg.]−19.0566037735849)×(t_TiO₂[λ]−0.0288364779874214))+(−526.442885320397)×((t_Si₂[λ]−0.408490566037736)×(t_TiO₂[λ]−0.0288364779874214))+(−298.795469471375)×((t_TiO₂[λ]−0.0288364779874214)×(t_TiO₂[λ]−0.0288364779874214)−0.000424904078161465)+(−50.1009078768921)×((Si_psi[deg.]−19.0566037735849)×(t_SiO₂[λ]−0.160062893081761))+1038.08065133921×((t_Si₂[λ])−0.408490566037736)×(t_SiO₂[λ]−0.160062893081761))+(−1286.74436136556)×((t_TiO₂[λ]−0.0288364779874214)×(t_SiO₂[λ]−0.160062893081761))+4158.8148931551×((t_SiO₂[λ]−0.160062893081761)×(t_SiO₂[λ]−0.160062893081761)−0.00134527906332819)  Equation 13

Si_psi[deg.], t_SiO₂[λ], t_TiO₂[λ], and t_Si₂[λ] each are preferably avalue within a range with which y in the equation 13 is less than orequal to about −70. With this configuration, the phase of a higher-ordermode is further reliably set to less than or equal to about −70 [deg.].Therefore, higher-order modes are further reliably and effectivelyreduced.

In the configuration having a multilayer substrate 39 similar to that ofthe fourth preferred embodiment, the plane orientation of the siliconsubstrate 2 was set to (110), and the phase of a higher-order mode wasmeasured while the parameters were changed. The design parameters andthe variable ranges of the design parameters were similar to those whenthe equation 13 was derived except the angle α.

-   -   α₁₁₀: changed in increments of 10° in the range greater than or        equal to 0° and less than or equal to 90°

The phase of a higher-order mode was measured while the parameters werechanged as described above. Thus, an equation 14 that was a relationalexpression between the parameters and the phase of a higher-order modewas derived. In the equation 14, Si_psi [deg.] is the angle α₁₁₀.

y[deg.]=(−66.0681190864303)+(−0.0323391014318074)×(Si_psi[deg.]−35.9295352323838)+0.997507104337367×(t_Si₂[λ]−0.394527736131933)+155.754971155735×(t_TiO₂[λ]−0.0378860569715143)+(−27.4736558331949)×(t_SiO₂[λ]−0.149887556221888)+0.00791197424152189×((Si_psi[deg.]−35.9295352323838)×(Si_psi[deg.]−35.9295352323838)−256.81962242267)+0.212504000649305×((Si_psi[deg.]−35.9295352323838)×(t_Si₂[λ]−0.394527736131933))+88.6294722534935×((t_Si₂[λ]−0.394527736131933)×(t_Si₂[λ]−0.394527736131933)−0.0392241772666888)+0.636412965393882×((Si_psi[deg.]−35.9295352323838)×(t_TiO₂[λ]−0.0378860569715143))+157.120610191294×((t_Si₂[λ]−0.394527736131933)×(t_TiO₂[λ]−0.0378860569715143))+544.188337615988×((t_TiO₂[λ]−0.0378860569715143)×(t_TiO₂[λ]−0.0378860569715143)−0.000522930045472021)+0.408031229502175×((Si_psi[deg.]−35.9295352323838)×(t_SiO₂[λ]−0.149887556221888))+(−46.0736528123303)×((t_Si₂[λ]−0.394527736131933)×(t_SiO ₂[R]−0.149887556221888))+(−1322.9465191866)×((t_TiO₂[λ]−0.0378860569715143)×(t_SiO₂[λ]−0.149887556221888))+359.098768522305×((t_SiO₂[λ])−0.149887556221888)×(t_SiO₂[λ]−0.149887556221888)−0.00163979245384803)  Equation 14

Si_psi[deg.], t_SiO₂[λ], t_TiO₂[λ], and t_Si₂[λ] each are preferably avalue within a range with which y in the equation 14 is less than orequal to about −70. With this configuration, the phase of a higher-ordermode is further reliably set to less than or equal to about −70 [deg.].Therefore, higher-order modes are further reliably and effectivelyreduced.

In the configuration of the fourth preferred embodiment, the planeorientation of the silicon substrate 2 was set to (111), and the phaseof a higher-order mode was measured while the parameters were changed.The design parameters and the variable ranges of the design parameterswere similar to those when the equation 13 was derived except the angleα.

-   -   α₁₁₁: changed in increments of 5° in the range greater than or        equal to 0° and less than or equal to 60°

The phase of a higher-order mode was measured while the parameters werechanged as described above. Thus, an equation 15 that was a relationalexpression between the parameters and the phase of a higher-order modewas derived. In the equation 15, Si_psi [deg.] is the angle α₁₁₁.

y[deg.]=(−69.4030815485713)+(−0.269371737613053)×(Si_psi[deg.]−33.8730694980695)+(−4.68577968707475)×(t_Si₂[λ]−0.440745656370656)+176.177168052005×(t_LT[λ]−0.168858590733587)+73.1412385401181×(t_TiO₂[λ]−0.0300916988416992)+(−12.3739066281753)×(t_SiO₂[λ]−0.154983108108108)+0.00777703537774127C((Si_psi[deg.]−33.8730694980695)×(Si_psi[deg.]−33.8730694980695)−508.406668570442)+0.121265989497045×((Si_psi[deg.]−33.8730694980695)×(t_Si₂[λ]−0.440745656370656))+17.8168374568741×((t_Si₂[λ]−0.440745656370656)×(t_Si₂[λ]−0.440745656370656)−0.0493816592475423)+1.34130425597794×((Si_psi[deg.]−33.8730694980695)×(t_LT[λ]−0.168858590733587))+(−5.11032690396319)×((t_Si₂[λ]−0.440745656370656)×(t_LT[λ]−0.168858590733587))+2.48016332864734×((Si_psi[deg.]−33.8730694980695)×(t_TiO₂[λ]−0.0300916988416992))+77.8145877606436×((t_Si₂[λ]−0.440745656370656)×(t_TiO₂[λ]−0.0300916988416992))+2112.87481803881×((t_LT[λ]−0.168858590733587)×(t_TiO₂[λ]−0.0300916988416992))+196.040518466468×((t_TiO₂[λ]−0.0300916988416992)×(t_TiO₂[λ]−0.0300916988416992)−0.000562395066225887)+(−0.969575065396993)×((Si_psi[deg.]−33.8730694980695)×(t_SiO₂[λ]−0.154983108108108))+(−138.70694337489)×((t_Si₂[λ]−0.440745656370656)×(t_SiO₂[λ]−0.154983108108108))+(−1100.04408119143)×((t_LT[λ]−0.168858590733587)×(t_SiO₂[λ]−0.154983108108108))+74.9944030678128×((t_TiO₂[λ]−0.0300916988416992)×(t_SiO₂[λ]−0.154983108108108))+117.812778429437×((t_SiO₂[λ]−0.154983108108108)×(t_SiO₂[λ]−0.154983108108108)−0.00117057162586093)  Equation 15

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], t_TiO₂[λ], and t_Si₂[λ] each arepreferably a value within a range with which y in the equation 15 isless than or equal to about −70. With this configuration, the phase of ahigher-order mode is further reliably set to less than or equal to about−70[deg.]. Therefore, higher-order modes are further reliably andeffectively reduced.

In addition, conditions in which the phase of a higher-order mode wasreduced to less than or equal to about −80[deg.] were obtained in eachof a case where the plane orientation of the silicon substrate 2 was(110) and a case where the plane orientation of the silicon substrate 2was (111). More specifically, to obtain the conditions, an equation 16and an equation 17 were derived while the parameters were changed withinthe range in which the phase of a higher-order mode was greater than orequal to −90[deg.] and less than or equal to about −70 [deg.].

When the plane orientation of the silicon substrate 2 is (110), theequation 16 was derived as described above.

y[deg.]=(−77.5229944626225)+(−0.0245637365901893)×(Si_psi[deg.]−34.5205479452055)+(−1.18326432300356)×(t_Si₂[λ]−0.408904109589041)+131.275052857081×(t_TiO₂[λ]−0.0160045662100456)+(−18.9167659640434)×(t_SiO₂[λ]−0.150228310502283)+0.00233031321934999×((Si_psi[deg.]−34.5205479452055)×(Si_psi[deg.]−34.5205479452055)−75.9116782385689)+(−0.0809397438263331)×((Si_psi[deg.]−34.5205479452055)×(t_Si₂[λ]−0.408904109589041))+43.1653284334043×((t_Si₂[λ]−0.408904109589041)×(t_Si₂[λ]−0.408904109589041)−0.0169869268780884)+(−0.255179621676294)×((Si_psi[deg.]−34.5205479452055)×(t_TiO₂[λ])−0.0160045662100456))+(−101.438420329361)×((t_Si₂[λ]−0.408904109589041)×(t_TiO₂[λ]−0.0160045662100456))+(−834.553397134957)×((t_TiO₂[λ]−0.0160045662100456)×(t_TiO₂[λ]−0.0160045662100456)−0.000126844728008173)+0.935627393438751×((Si_psi[deg.]−34.5205479452055)×(t_SiO₂[λ])−0.150228310502283))+(−21.1152350576513)×((t_Si₂[λ]−0.408904109589041)×(t_SiO₂[λ]−0.150228310502283))+(−41.8110780477077)×((t_TiO₂[λ]−0.0160045662100456)×(t_SiO₂[λ]−0.150228310502283))+263.939639423742×((t_SiO₂[λ]−0.150228310502283)×(t_SiO₂[λ]−0.150228310502283)−0.00160953691541044)  Equation 16

Si_psi[deg.], t_SiO₂[λ], t_TiO₂[λ], and t_Si₂[λ] each are preferably avalue within a range with which y in the equation 16 is less than orequal to about −80. With this configuration, the phase of a higher-ordermode is further reliably set to less than or equal to about −80[deg.].Therefore, higher-order modes are further reliably and further reduced.

When the plane orientation of the silicon substrate 2 is (111), theequation 17 was derived as described above.

y[deg.]=(−78.9096176038851)+0.034903516437231×(Si_psi[deg.]−45.8453473132372)+(−2.02533584474356)×(t_Si₂[λ]−0.421068152031454)+81.4363661536651×(t_LT[λ]−0.156225425950199)+71.7903756668229×(t_TiO₂[λ]−0.0334862385321101)+(−3.53261283561548)×(t_SiO₂[λ]−0.149606815203146)+0.00246176329064131×((Si_psi[deg.]−45.8453473132372)×(Si_psi[deg.]−45.8453473132372)−143.191023568758)+(−0.0293712406566626)×((Si_psi[deg.]−45.8453473132372)×(t_Si₂ [AX]−0.421068152031454))+0.972512275661977×((t_Si₂[λ]−0.421068152031454)×(t_Si₂[λ]−0.421068152031454)−0.0430997109516307)+0.30260322985253×((Si_psi[deg.]−45.8453473132372)×(t_LT[λ]−0.156225425950199))+(−23.2249863870645)×((t_Si₂[λ]−0.421068152031454)×(t_LT[λ]−0.156225425950199))+0.513496313876766×((Si_psi[deg.]−45.8453473132372)×(t_TiO₂[λ]−0.0334862385321101))+(−143.065209527507)×((t_Si₂[λ]−0.421068152031454)×(t_TiO₂[λ]−0.0334862385321101))+(−324.329178613173)×((t_LT[λ]−0.156225425950199)×(t_TiO₂[λ]−0.0334862385321101))+(−98.4001544132927)×((t_TiO₂[λ]−0.0334862385321101)×(t_TiO₂[λ]−0.0334862385321101)−0.000480867110753061)+(−1.15889670547368)×((Si_psi[deg.]−45.8453473132372)×(t_SiO₂[λ]−0.149606815203146))+(−50.3263112114924)×((t_Si₂[λ]−0.421068152031454)×(t_SiO₂[λ]−0.149606815203146))+(−209.199256641353)×((t_LT[λ]−0.156225425950199)×(t_SiO₂[λ]−0.149606815203146))+90.23187502294813×((t_TiO₂[λ]−0.0334862385321101)×(t_SiO₂[λ]−0.149606815203146))+347.448658314796×((t_SiO₂[λ]−0.149606815203146)×(t_SiO₂[λ]−0.149606815203146)−0.00114008131659363)  Equation 17

Si_psi[deg.], t_LT[λ], t_SiO₂[λ], t_TiO₂[λ], and t_Si₂[λ] each arepreferably a value within a range with which y in the equation 17 isless than or equal to about −80. With this configuration, the phase of ahigher-order mode is further reliably set to less than or equal to about−80[deg.]. Therefore, higher-order modes are further reliably andfurther reduced.

In the above description, an example in which the piezoelectric layer 7is a lithium tantalate layer has been described. Hereinafter, an examplein which the piezoelectric layer 7 is a lithium niobate layer will bedescribed with reference to FIG. 17 .

A fifth preferred embodiment of the present invention differs from thesecond preferred embodiment in that the piezoelectric layer 7 is alithium niobate layer. Other than the above points, the acoustic wavedevice according to the fifth preferred embodiment has a similarconfiguration to the acoustic wave device according to the secondpreferred embodiment.

Here, the phase characteristics of the acoustic wave device having theconfiguration of the fifth preferred embodiment were measured. Thedesign parameters of the acoustic wave device are as follows.

-   -   Silicon Substrate 2: Material monocrystal silicon, Plane        orientation (111), Euler angles ((φ, θ, ψ) (−45°, −54.7°, 30°),        and Thickness 20 μm    -   Polysilicon Layer 3: Material polysilicon, and Thickness 1 μm    -   Silicon Oxide Layer 5: Material SiO₂, and Thickness 300 nm    -   Silicon Nitride Layer 26: Material SiN, and Thickness 30 nm    -   Piezoelectric Layer 7: Material LiNbO₃, Cut Angle 40° Y, Euler        angles ((φ, θ, ψ) (0°, 130°, 0°), and Thickness 300 nm    -   Layer Configuration of Interdigital Transducer Electrode 8:        Material Ti, AlCu, and Ti from the piezoelectric layer 7 side,        the content of Cu in AlCu is 1 wt %, and Thickness 10 nm, 100        nm, and 4 nm from the piezoelectric layer 7 side    -   Duty Ratio of Interdigital Transducer Electrode 8: 0.5    -   Wave Length λ of Interdigital Transducer Electrode 8: 2 μm    -   Protective Film 29: Material SiO₂, and Thickness 30 nm

Here, the present preferred embodiment and a second comparative exampleare compared with each other to demonstrate that higher-order modes arereduced in a wide band according to the present preferred embodiment.The second comparative example differs from the present preferredembodiment in that, in the multilayer substrate, a silicon nitride layeris laminated instead of the polysilicon layer.

FIG. 23 is a graph that shows the phase characteristics of the acousticwave device according to the fifth preferred embodiment and the phasecharacteristics of an acoustic wave device according to the secondcomparative example.

As shown in FIG. 23 , it appears that higher-order modes are reduced ina wider band in the fifth preferred embodiment than in the secondcomparative example. In this way, in the fifth preferred embodiment aswell, as in the case of the second preferred embodiment, higher-ordermodes are reduced in a wide band. In addition, as shown in FIG. 23 , itappears that the band of a main mode is expanded.

While preferred embodiments of the present invention have been describedabove, it is to be understood that variations and modifications will beapparent to those skilled in the art without departing from the scopeand spirit of the present invention. The scope of the present invention,therefore, is to be determined solely by the following claims.

1: An acoustic wave device comprising: a silicon substrate; apolysilicon layer provided on the silicon substrate; a silicon oxidelayer directly or indirectly provided on the polysilicon layer; apiezoelectric layer directly or indirectly provided on the silicon oxidelayer; and an interdigital transducer electrode provided on thepiezoelectric layer; wherein a plane orientation of the siliconsubstrate is any one of (100), (110), and (111); and where a wave lengththat is defined by an electrode finger pitch of the interdigitaltransducer electrode is λ, a thickness of the piezoelectric layer isless than or equal to about 1λ. 2: The acoustic wave device according toclaim 1, further comprising a silicon nitride layer provided between thesilicon oxide layer and the piezoelectric layer. 3: The acoustic wavedevice according to claim 1, further comprising a silicon nitride layerprovided between the polysilicon layer and the silicon oxide layer. 4:The acoustic wave device according to claim 1, further comprising atitanium oxide layer provided between the silicon oxide layer and thepiezoelectric layer. 5: The acoustic wave device according to claim 1,wherein the piezoelectric layer is a lithium tantalate layer; thepiezoelectric layer has crystallographic axes (X_(P), Y_(P), Z_(P)); theplane orientation of the silicon substrate is (100); in the siliconsubstrate of which the plane orientation is (100), a directional vectorobtained by projecting the Z_(P)-axis onto a (100) plane of the siliconsubstrate is k₁₀₀, and an angle between the directional vector k₁₀₀ anda [001] direction of silicon that is a component of the siliconsubstrate is an angle of α₁₀₀; and where the angle α₁₀₀ is Si_psi[deg.],a thickness of the piezoelectric layer is t_LT[λ], a thickness of thesilicon oxide layer is t_SiO₂[λ], and a thickness of the polysiliconlayer is t_Si₂[λ], the Si_psi[deg.], the t_LT[λ], the t_SiO₂[λ], and thet_Si₂[λ] each are a value within a range with which y in equation 1 isless than or equal to about −70:y[deg.]=(−72.1492542241195)+0.627588217157224×(Si_psi[deg]−21.7083333333333)+(−1.93347870945237)×(t_Si₂[λ]−0.4525)+72.3846086764674×(t_LT[λ]−0.160833333333333)+(−67.3219584197057)×(t_SiO₂[λ]−0.16625)+0.0000655654050315201×((Si_psi[deg.]−21.7083333333333)×(Si_psi[deg.]−21.7083333333333)−25.2065972222222)+(−2.34857364418332)×((Si_psi[deg.]−21.7083333333333)×((t_Si₂[λ]−0.4525))+37.0048979126418×((t_Si ₂[λ]−0.4525)×((t_Si₂[λ]−0.4525)−0.0360354166666667)+7.0771357128953×((Si_psi[deg.]−21.7083333333333)×((t_LT[λ]−0.160833333333333))+(−10.057857939681)×((t_Si₂[λ]−0.4525)×(t_LT[λ]−0.160833333333333))+1.50716777611893×((Si_psi[deg.]−21.7083333333333)×(t_SiO₂[λ]−0.16625))+426.86632497558×(((t_Si ₂[λ]−0.4525)×((t_SiO₂[λ])−0.16625))+925.280868396996×((t_LT[λ]−0.160833333333333)×(t_SiO₂[λ]−0.16625))+988.798729044457×((t_SiO ₂[λ]−0.16625)×(t_SiO₂[λ]−0.16625)−0.000871354166666668).  Equation 1 6: The acoustic wavedevice according to claim 1, wherein the piezoelectric layer is alithium tantalate layer; the piezoelectric layer has crystallographicaxes (X_(P), Y_(P), Z_(P)); the plane orientation of the siliconsubstrate is (110); in the silicon substrate of which the planeorientation is (110), a directional vector obtained by projecting theZ_(P)-axis onto a (110) plane of the silicon substrate is k₁₁₀, and anangle between the directional vector k₁₁₀ and a [001] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₀; and where the angle α₁₁₀ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon oxide layeris t_SiO₂[λ], and a thickness of the polysilicon layer is t_Si₂[λ], theSi_psi[deg.], the t_LT[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are avalue within a range with which y in equation 2 is less than or equal toabout −70:y[deg.]=(78.1876454049157)+(−0.182894322081067)×(Si_psi[deg.]−28.1088082901554)+6.18390256271178×(t_Si₂[λ]−0.39961139896373)+116.669335737855×(t_LT[λ]−0.169948186528498)+10.3573467893808×(t_SiO₂[λ]−0.144041450777202)+0.0110735958981267×((Si_psi[deg.]−28.1088082901554)×(Si_psi[deg.]−28.1088082901554)−189.946709978791)+(−0.246858144090431)×((Si_psi[deg.]−28.1088082901554)×(t_Si₂[λ]−0.39961139896373)+22.031016276383×((t_Si₂[λ]−0.39961139896373)×(t_Si₂[λ]−0.39961139896373)−0.0484389681602191)+(−X.0545756011518778)×((Si_psi[deg.]−28.1088082901554)×(t_LT[λ]−0.169948186528498))+(−32.427969747408)×((t_Si₂[λ]−0.39961139896373)×((t_LT[λ]−0.169948186528498))+(−2.62164982026802)×((Si_psi[deg.]−28.1088082901554)×(t_SiO₂[λ]−0.144041450777202))+(−112.759047075747)×((t_Si₂[λ]−0.39961139896373)×((t_SiO₂[λ]−0.144041450777202))+(−604.832727678973)×((t_LT[λ]−0.169948186528498)×((t_SiO₂[λ]−0.144041450777202))+326.415587634024×((t_SiO₂[λ]−0.144041450777202)×((t_SiO₂[λ]−0.144041450777202)−0.00120154232328385).  Equation 2 7: Theacoustic wave device according to claim 1, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (111); in the silicon substrate of which theplane orientation is (111), a directional vector obtained by projectingthe Z_(P)-axis onto a (111) plane of the silicon substrate is k₁₁₁, andan angle between the directional vector k₁₁₁ and a [11-2] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₁; and where the angle α₁₁₁ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon oxide layeris t_SiO₂[λ], and a thickness of the polysilicon layer is t_Si₂[λ], theSi_psi[deg.], the t_LT[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are avalue within a range with which y in equation 3 is less than or equal toabout −70:y[deg.]=(77.9109394183719)+(−0.0492368384201428)×(Si_psi[deg.]−45.2068126520681)+0.525124426223863×(t_Si₂[λ]−0.426216545012165)+117.400884406373×(t_LT[λ]−0.174330900243311)+(−2.62484877324049)×(t_SiO₂[λ]−0.15139902676399)+0.00307563131201403×((Si_psi[deg.]−45.2068126520681)×(Si_psi[deg.]−45.2068126520681)−182.925598356629)+(−0.0261801752592506)×((Si_psi[deg.]−45.2068126520681)×(t_Si₂[λ]−0.426216545012165))+23.8987529211434×((t_Si₂[λ]−0.426216545012165)×(t_Si₂[λ]−0.426216545012165)−0.0481296027432942)+1.52616542281399×((Si_psi[deg.]−45.2068126520681))×(t_LT[λ]−0.174330900243311))+(−129.002027283367)×((t_Si₂[λ]−0.426216545012165)×((t_LT[λ]−0.174330900243311))+(−1.22761778451819)×((Si_psi[deg.]−45.2068126520681)×((t_SiO₂[λ]−0.15139902676399))+(−42.6041784800926)×((t_Si₂[λ]−0.426216545012165)×(t_SiO₂[λ]−0.15139902676399))+(−468.84116493048)×((t_LT[λ]−0.174330900243311)×(t_SiO₂[λ]−0.15139902676399))+(−8.20635607220859)×((t_SiO₂[λ]−0.15139902676399)×(t_SiO₂[λ]−0.15139902676399)−0.0012830183932134).  Equation 3 8: The acousticwave device according to claim 1, wherein the piezoelectric layer is alithium tantalate layer; the piezoelectric layer has crystallographicaxes (X_(P), Y_(P), Z_(P)); the plane orientation of the siliconsubstrate is (100); in the silicon substrate of which the planeorientation is (100), a directional vector obtained by projecting theZ_(P)-axis onto a (100) plane of the silicon substrate is k₁₀₀, and anangle between the directional vector k₁₀₀ and a [001] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₀₀; and where the angle α₁₀₀ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon oxide layeris t_SiO₂[λ], and a thickness of the polysilicon layer is t_Si₂[λ], theSi_psi[deg.], the t_LT[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are avalue within a range with which y in equation 4 is less than or equal toabout −80:y[deg.]=(−75.3156232479379)+0.63547968892276×(Si_psi[deg]−20.9090909090909)+(−2.02838142816204)×(t_Si₂[λ]−0.439772727272727)+90.1874317877843×(t_LT[λ]−0.151136363636364)+(−71.2997621594781)×(t_SiO₂[λ]−0.171590909090909)+0.108397383766316×((Si_psi[deg.]−20.9090909090909)×(Si_psi[deg.]−20.9090909090909)−13.9462809917355)+(−3.76982864951476)×((Si_psi[deg.]−20.9090909090909)×(t_Si₂[λ]−0.439772727272727))+37.3378798744213×((t_Si₂[λ]−0.439772727272727)×(t_Si₂[λ]−0.439772727272727)−0.0358613119834711)+(−23.7942425679855)×((Si_psi[deg.]−20.9090909090909)×(t_SiO₂[λ]−0.171590909090909))+462.018905986831×((t_Si₂[λ])−0.439772727272727)×(t_SiO₂[λ]−0.171590909090909))+1223.13016730739×(((t_SiO₂[λ]−0.171590909090909)×(t_SiO₂[λ]−0.171590909090909)−0.000641787190082645).  Equation 4 9: Theacoustic wave device according to claim 1, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (110); in the silicon substrate of which theplane orientation is (110), a directional vector obtained by projectingthe Z_(P)-axis onto a (110) plane of the silicon substrate is k₁₁₀, andan angle between the directional vector k₁₁₀ and a [001] direction ofsilicon that is a component of the silicon substrate is an angle of α₁₁₀and where the angle α₁₁₀ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon oxide layeris t_SiO₂[λ], and a thickness of the polysilicon layer is t_Si₂[λ], theSi_psi[deg.], the t_LT[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are avalue within a range with which y in equation 5 is less than or equal toabout −80:y[deg.]=(81.4138269086073)+(−0.100532115186538)×(Si_psi[deg.]−29.1379310344828)+0.845708574223377×(t_Si₂[λ]−0.385689655172414)+87.6682874459356×(t_LT[λ]−0.166724137931034)+(−0.137780433371857)×(t_SiO₂[λ]−0.145)+0.00337749443465239×((Si_psi[deg.]−29.1379310344828)×(Si_psi[deg.]−29.1379310344828)−127.877526753864)+(−0.116548121456389)×((Si_psi[deg.]−29.1379310344828)×(t_Si₂[λ]−0.385689655172414))+11.8893452691356×((t_Si₂[λ]−0.385689655172414)×(t_Si₂[λ]−0.385689655172414)−0.0448900416171225)+0.333200244545922×((Si_psi[deg.]−29.1379310344828)×(t_LT[λ]−0.166724137931034))+55.2630600466406×((t_Si₂[λ]−0.385689655172414)×(t_LT[λ]−0.166724137931034))+(−0.296582437395607)×((Si_psi[deg.]−29.1379310344828)×(t_SiO₂[λ]−0.145))+(−67.4578937630203)×((t_Si ₂[λ]−0.385689655172414)×(t_SiO₂[λ]−0.145))+(−376.292315976729)×((t_LT[λ])−0.166724137931034)×(t_SiO₂[λ]−0.145))+48.6290874437329×((t_SiO ₂[λ]−0.145)×((t_SiO₂[λ]−0.145)−0.00120775862068966).  Equation 5 10: The acoustic wavedevice according to claim 1, wherein the piezoelectric layer is alithium tantalate layer; the piezoelectric layer has crystallographicaxes (X_(P), Y_(P), Z_(P)); the plane orientation of the siliconsubstrate is (111); in the silicon substrate of which the planeorientation is (111), a directional vector obtained by projecting theZ_(P)-axis onto a (111) plane of the silicon substrate is k₁₁₁, and anangle between the directional vector k₁₁₁ and a [11-2] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₁; and where the angle α₁₁₁ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon oxide layeris t_SiO₂[λ], and a thickness of the polysilicon layer is t_Si₂[λ], theSi_psi[deg.], the t_LT[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are avalue within a range with which y in equation 6 is less than or equal toabout −80:y[deg.]=(−79.8924944284088)+0.033261334588906×(Si_psi[deg.]−39.3173431734317)+3.93783296791666×(t_Si₂[λ]−0.416974169741698)+80.6680077909648×(t_LT[λ]−0.17140221402214)+13.2276438709535×(t_SiO₂[λ]−0.148431734317343)+(−0.00907764275073328)×((Si_psi[deg.]−39.3173431734317)×(Si_psi[deg.]−39.3173431734317)−21.2129464468077)+0.000540095694459618×((Si_psi[deg.]−39.3173431734317)×(t_Si₂[λ]−0.416974169741698))+5.79698263968963×((t_Si₂[λ]−0.416974169741698)×(t_Si₂[λ]−0.416974169741698)−0.0400439808826132)+(−0.136650035849863)×((Si_psi[deg.]−39.3173431734317)×(t_LT[λ]−0.17140221402214))+(−20.3328823416631)×((t_Si₂[λ]−0.416974169741698)×(t_LT[λ]−0.17140221402214))+(−2.22480760136672)×((Si_psi[deg.]−39.3173431734317)×(t_SiO₂[λ]−0.148431734317343))+(−13.0975601885972)×((t_Si₂[λ]−0.416974169741698)×(t_SiO₂[λ]−0.148431734317343))+(−511.743077543129)×((t_LT[λ]−0.17140221402214)×(t_SiO₂[λ]−0.148431734317343))+137.213612130809×((t_SiO₂[λ]−0.148431734317343)×(t_SiO₂[λ]−0.148431734317343)−0.00135593537669694).  Equation 6 11: Theacoustic wave device according to claim 2, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (100); in the silicon substrate of which theplane orientation is (100), a directional vector obtained by projectingthe Z_(P)-axis onto a (100) plane of the silicon substrate is k₁₀₀, andan angle between the directional vector k₁₀₀ and a [001] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₀₀; and where the angle α₁₀₀ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon nitride layeris t_SiN[λ], a thickness of the silicon oxide layer is t_SiO₂[λ], and athickness of the polysilicon layer is t_Si₂[λ], the Si_psi[deg.], thet_LT[λ], the t_SiN[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are a valuewithin a range with which y in equation 7 is less than or equal to about−70:y[deg.]=(−67.7782730918073)+0.0667732718475358×(Si_psi[deg.]−25.6259314456036)+(−6.71256568714434)×(t_Si₂[λ]−0.426192250372578)+177.355083873051×(t_LT[λ]−0.16602086438151)+(−64.7093491078986)×(t_SiN[λ]−0.0465201192250378)+1.0890884781807×(t_SiO₂[λ]−0.155793591654245)+0.000179985859065592×(Si_psi[deg.]−25.6259314456036)×(Si_psi[deg.]−25.6259314456036)−130.38317256758)+(−0.329348427439478)×((Si_psi[deg.]−25.6259314456036)×(t_Si₂[λ]−0.426192250372578))+(−33.1084698932093)×((t_Si₂[λ]−0.426192250372578)×(t_Si₂[λ]−0.426192250372578)−0.0504801359160987)+1.52146775761601×((Si_psi[deg.]−25.6259314456036)×(t_LT[λ]−0.16602086438151))+14.59741625744683×((t_Si₂[λ]−0.426192250372578)×(t_LT[λ]−0.16602086438151))+0×((t_LT[λ]−0.16602086438151)×(t_LT[λ]−0.16602086438151)−0.000544375123544922)+(−4.94058423048505)×((Si_psi[deg.]−25.6259314456036)×((t_SiN[λ]−0.0465201192250378))+138.799085167873×((t_Si×[λ]−0.426192250372578)×(t_SiN[λ]−0.0465201192250378))+1746.7447498235×((t_LT[λ]−0.16602086438151)×(t_SiN[λ]−0.0465201192250378))+2167.04168685901×((t_SiN[λ]−0.0465201192250378)×(t_SiN[λ]−0.0465201192250378)−0.000465274930537198)+(−0.931372972560935)×((Si_psi[deg.]−25.6259314456036)×(t_SiO₂[λ]−0.155793591654245))+(−79.4377446578721)×((t_Si₂[λ]−0.426192250372578)×(t_SiO₂[λ]−0.155793591654245))+(−86.9697272546991)×((t_LT[λ]−0.16602086438151)×(t_SiO₂[λ]−0.155793591654245))+1966.46522796354×((t_SiN[λ]−0.0465201192250378)×(t_SiO₂[λ]−0.155793591654245))+169.040605778099×((t_SiO₂[λ]−0.155793591654245)×(t_SiO₂[λ]−0.155793591654245)−0.00164210493657841).  Equation 7 12: Theacoustic wave device according to claim 2, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (110); in the silicon substrate of which theplane orientation is (110), a directional vector obtained by projectingthe Z_(P)-axis onto a (110) plane of the silicon substrate is k₁₁₀, andan angle between the directional vector k₁₁₀ and a [001] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₀; and where the angle α₁₁₀ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon nitride layeris t_SiN[λ], a thickness of the silicon oxide layer is t_SiO₂[λ], and athickness of the polysilicon layer is t_Si₂[λ], the Si_psi[deg.], thet_LT[λ], the t_SiN[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are a valuewithin a range with which y in equation 8 is less than or equal to about−70:y[deg.]=(−75.0174122935603)+(−0.00810936153116664)×(Si_psi[deg.]−42.0340722495895)+1.98135617767495×(t_Si₂[λ]−0.385026683087027)+143.173790020328×(t_LT[λ]−0.17306034482757)+16.4148627328736×(t_SiN[λ]−0.04207922824302)+50.4122771861205×(t_SiO₂AR)−0.144909688013139)+0.00619821963137332×((Si_psi[deg.]−42.0340722495895)×(Si_psi[deg.]−42.0340722495895)−514.232829229589)+0.020323078287526×((Si_psi[deg.]−42.0340722495895)×(t_Si₂[λ]−0.385026683087027))+1.15443318031007×((t_Si₂[λ]−0.385026683087027)×(t_Si₂[λ]−0.385026683087027)−0.0477966331139576)+0.472662465737381×((Si_psi[deg.]−42.0340722495895)×(t_LT[λ]−0.17306034482757))+(−105.2996012677)×((t_Si₂[λ]−0.385026683087027)×(t_LT[λ]−0.17306034482757))+(−1.29517116632701)×((Si_psi[deg.]−42.0340722495895)×(t_SiN[λ]−0.04207922824302))+(−26.1801037669841)×((t_Si₂[λ]−0.385026683087027)×(t_SiN[λ])−0.04207922824302))+168.1334353773×((t_LT[λ]−0.17306034482757)×(t_SiN[λ]−0.04207922824302))+2120.76431830662×((t_SiN[λ]−0.04207922824302)×(t_SiN[λ]−0.04207922824302)−0.000508197335364991)+(−0.687562974959064)×((Si_psi[deg.]−42.0340722495895)×(t_SiO₂[λ]−0.144909688013139))+15.3482271106745×((t_Si₂[λ]−0.385026683087027)×(t_SiO₂[λ]−0.144909688013139))+(−358.720795782422)×((t_LT[λ]−0.17306034482757)×(t_SiO₂[λ]−0.144909688013139))+1062.30534015379×((t_SiN[λ]−0.04207922824302)×(t_SiO₂[λ]−0.144909688013139))+248.937429294479×((t_SiO₂[λ]−0.144909688013139)×(t_SiO₂[λ]−0.144909688013139)−0.00162330875671721).  Equation 8 13: Theacoustic wave device according to claim 2, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (111); in the silicon substrate of which theplane orientation is (111), a directional vector obtained by projectingthe Z_(P)-axis onto a (111) plane of the silicon substrate is k₁₁₁, andan angle between the directional vector k₁₁₁ and a [11-2] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₁; and where the angle α₁₁₁ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon nitride layeris t_SiN[λ], a thickness of the silicon oxide layer is t_SiO₂[λ], and athickness of the polysilicon layer is t_Si₂[λ], the Si_psi[deg.], thet_LT[λ], the t_SiN[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are a valuewithin a range with which y in equation 9 is less than or equal to about−70:y[deg.]=(−77.5405307874512)+0.00496521862619995×(Si_psi[deg.]−44.3479880774963)+(−3.07514699616305)×(t_Si₂[λ]−0.395628415300543)+115.725430166886×(t_LT[λ]−0.173919523099848)+75.6109484741613×(t_SiN[λ]−0.0387729756582212)+29.9143205043822×(t_Si₂[λ]−0.145404868355688)+0.00452378218877289×((Si_psi[deg.]−44.3479880774963)×(Si_psi[deg.]−44.3479880774963)−147.519490487682)+(−0.127045459018856)×((Si_psi[deg.]−44.3479880774963)×(t_Si₂[λ]−0.395628415300543))+10.135015813019×((t_Si₂[λ]−0.395628415300543)×(t_Si₂[λ]−0.395628415300543)−0.0544331992323139)+0.267609205446981×((Si_psi[deg.]−44.3479880774963)×(t_LT[λ]−0.173919523099848))+(−151.966315117959)×((t_Si₂[λ]−0.395628415300543)×(t_LT[λ]−0.173919523099848))+1.1818941610908×((Si_psi[deg.]−44.3479880774963)×(t_SiN[λ]−0.0387729756582212))+(−19.0228093275549)×((t_Si₂[λ]−0.395628415300543)×(t_SiN[λ]−0.0387729756582212))+25.2693219567039×((t_LT[λ]−0.173919523099848)×(t_SiN[λ])−0.0387729756582212))+1545.52112794945×((t_SiN[λ]−0.0387729756582212)×(t_SiN[λ]−0.0387729756582212)−0.000519520243356094)+(−0.39161225199813)×((Si_psi[deg.]−44.3479880774963)×(t_SiO₂[λ]−0.145404868355688))+22.0391330835907×((t_Si₂[λ]−0.395628415300543)×(t_SiO₂[λ]−0.145404868355688))+(−297.764935637906)×((t_LT[λ]−0.173919523099848)×(t_SiO₂[λ]−0.145404868355688))+982.324171494675×((t_SiN[λ]−0.0387729756582212)×(t_SiO₂[λ]−0.145404868355688))+420.570041600812×((t_SiO₂[λ]−0.145404868355688)×(t_SiO₂[λ]−0.145404868355688)−0.00124068307615005).  Equation 9 14: Theacoustic wave device according to claim 2, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (100); in the silicon substrate of which theplane orientation is (100), a directional vector obtained by projectingthe Z_(P)-axis onto a (100) plane of the silicon substrate is k₁₀₀, andan angle between the directional vector k₁₀₀ and a [001] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₀₀; and where the angle α₁₀₀ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon nitride layeris t_SiN[λ], a thickness of the silicon oxide layer is t_SiO₂[λ], and athickness of the polysilicon layer is t_Si₂[λ], the Si_psi[deg.], thet_LT[λ], the t_SiN[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are a valuewithin a range with which y in equation 10 is less than or equal toabout −80:y[deg.]=(−78.3557914112162)+(−0.00785147182473267)×(Si_psi[deg.]−24.9802110817942)+(−1.32878861667394)×(t_Si₂[λ]−0.429221635883905)+(−41.7937386863014)×(t_LT[λ]−0.150923482849606)+35.6722090195008×(t_SiN[λ]−0.0500263852242746)+18.7743164986736×(t_SiO₂[λ]−0.145646437994723)+(−0.000765722206063909)×((Si_psi[deg.]−24.9802110817942)×(Si_psi[deg.]−24.9802110817942)−153.396706024045)+(−0.0463379291760545)×((Si_psi[deg.]−24.9802110817942)×(t_Si₂[λ]−0.429221635883905))+(−17.7293821535291)×((t_Si₂[λ]−0.429221635883905)×(t_Si₂[λ]−0.429221635883905)−0.0593208981070862)+(−1.3441873888418)×((Si_psi[deg.]−24.9802110817942)×(t_LT[λ]−0.150923482849606))+(−417.636233521175)×((t_Si₂[λ]−0.429221635883905)×(t_LT[λ]−0.150923482849606))+(−0.487351707638102)×((Si_psi[deg.]−24.9802110817942)×(t_SiN[λ]−0.0500263852242746))+(−25.3025544220714)×((t_Si₂[λ]−0.429221635883905)×(t_SiN[λ]−0.0500263852242746))+1666.3381560311×((t_LT[λ]−0.150923482849606)×(t_SiN[λ]−0.0500263852242746))+233.559062145034×((t_SiN[λ]−0.0500263852242746)×(t_SiN[λ]−0.0500263852242746)−0.000389115398806747)+(−0.148028298904273)×((Si_psi[deg.]−24.9802110817942)×(t_SiO₂[λ]−0.145646437994723))+(−63.9722673973965)×((t_Si×[λ]−0.429221635883905)×(t_SiO₂[λ]−0.145646437994723))+1197.10044921435×((t_LT[λ]−0.150923482849606)×(t_SiO₂[λ]−0.145646437994723))+450.45656510444×((t_SiN[λ]−0.0500263852242746)×(t_SiO₂[λ])−0.145646437994723))+(−37.7857111587959)×((t_SiO₂[λ]−0.145646437994723)×(t_SiO₂[λ]−0.145646437994723)−0.0017158749939084).  Equation 10 15: Theacoustic wave device according to claim 2, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (110); in the silicon substrate of which theplane orientation is (110), a directional vector obtained by projectingthe Z_(P)-axis onto a (110) plane of the silicon substrate is k₁₁₀, andan angle between the directional vector k₁₁₀ and a [001] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₀; and where the angle α₁₁₀ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon nitride layeris t_SiN[λ], a thickness of the silicon oxide layer is t_SiO₂[λ], and athickness of the polysilicon layer is t_Si₂[λ], the Si_psi[deg.], thet_LT[λ], the t_SiN[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are a valuewithin a range with which y in equation 11 is less than or equal toabout −80:y[deg.]=(−79.9409825800918)+0.00367175250563163×(Si_psi[deg.]−42.1225309675259)+(−1.19942177285592)×(t_Si₂[λ]−0.381570137261466)+91.8359644721651×(t_LT[λ]−0.164596585202533)+58.8431912005245×(t_SiN[λ]−0.0395698024774026)+16.9153289429696×(t_SiO₂[λ]−0.13875125544024)+0.00130491910714855×((Si_psi[deg.]−42.1225309675259)×(Si_psi[deg.]−42.1225309675259)−385.786124427809)+0.0745672315210127×((Si_psi[deg]−42.1225309675259)×(t_Si₂[λ]−0.381570137261466))+2.6699307571413×((t_Si ₂R1-0.381570137261466)×(t_Si₂[λ]−0.381570137261466)−0.0456605075514713)+(−0.377889849052574)×((Si_psi[deg.]−42.1225309675259)×(t_LT[λ]−0.164596585202533))+(−43.4148735553507)×((t_Si₂[λ]−0.381570137261466)×(t_LT[λ]−0.164596585202533))+(−0.378387168121428)×((Si_psi[deg.]−42.1225309675259)×(t_SiN[λ]−0.0395698024774026))+(−20.545088460627)×((t_Si₂[λ]−0.381570137261466)×(t_SiN[λ]−0.0395698024774026))+232.919108783203×((t_LT[λ]−0.164596585202533)×(t_SiN[λ]−0.0395698024774026))+840.791113736585×((t_SiN[λ]−0.0395698024774026)×(t_SiN[λ]−0.0395698024774026)−0.000464855104179262)+0.190837727117146×((Si_psi[deg.]−42.1225309675259)×(t_SiO₂[λ]−0.13875125544024))+0.695837098714372×((t_Si₂[λ]−0.381570137261466)×(t_SiO₂[λ]−0.13875125544024))+(−184.621593720628)×((t_LT[λ]−0.164596585202533)×(t_SiO₂[λ]−0.13875125544024))+607.033426600094×((t_SiN[λ]−0.0395698024774026)×(t_SiO₂[λ]−0.13875125544024))+142.721242732228×((t_SiO₂[λ]−0.13875125544024)×(t_SiO₂[λ]−0.13875125544024)−0.00152562510304392).  Equation 11 16: Theacoustic wave device according to claim 2, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (111); in the silicon substrate of which theplane orientation is (111), a directional vector obtained by projectingthe Z_(P)-axis onto a (111) plane of the silicon substrate is k₁₁₁, andan angle between the directional vector k₁₁₁ and a [11-2] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₁; and where the angle α₁₁₁ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the silicon nitride layeris t_SiN[λ], a thickness of the silicon oxide layer is t_SiO₂[λ], and athickness of the polysilicon layer is t_Si₂[λ], the Si_psi[deg.], thet_LT[λ], the t_SiN[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are a valuewithin a range with which y in equation 12 is less than or equal toabout −80:y[deg.]=(−79.8683540124538)+0.0118371753456289×(Si_psi[deg.]−44.4595052524568)+(−1.99138796522555)×(t_Si₂[λ]−0.413673331074209)+88.0775643151379×(t_LT[λ]−0.167705862419511)+46.4734172707698×(t_SiN[λ]−0.0351321585903086)+14.4134894109961×(t_SiO₂[λ]−0.142222975262623)+0.00167085752221365×((Si_psi[deg.]−44.4595052524568)×(Si_psi[deg.]−44.4595052524568)−128.282924729805)+(−0.0463012101323173)×((Si_psi[deg.]−44.4595052524568)×(t_Si₂[λ]−0.413673331074209))+4.58192618035487×((t_Si₂[λ]−0.413673331074209)×(t_Si₂[λ]−0.413673331074209)−0.05167257915661)+0.524887931323933×((Si_psi[deg]−44.4595052524568)×(t_LT[λ]−0.167705862419511))+(−71.7492658390069)×((t_Si₂[λ]−0.413673331074209)×(t_LT[λ]−0.167705862419511))+0.73863390529294×((Si_psi[deg.]−44.4595052524568)×(t_SiN[λ]−0.0351321585903086))+(−42.8957552454222)×((t_Si₂[λ]−0.413673331074209)×(t_SiN[λ]−0.0351321585903086))+(−411.839865840595)×((t_LT[λ]−0.167705862419511)×(t_SiN[λ]−0.0351321585903086))+982.235412331017×((t_SiN[λ]−0.0351321585903086)×(t_SiN[λ]−0.0351321585903086)−0.000477142818756284)+(−0.236509133242243)×((Si_psi[deg.]−44.4595052524568)×(t_SiO₂[λ]−0.142222975262623))+17.2370398551984×((t_Si₂[λ]−0.413673331074209)×(t_SiO₂[λ]−0.142222975262623))+(−469.933137492789)×((t_LT[λ]−0.167705862419511)×(t_SiO₂[λ]−0.142222975262623))+541.748349798792×((t_SiN[λ]−0.0351321585903086)×(t_SiO₂[λ]−0.142222975262623))+226.311489477246×((t_SiO₂[λ]−0.142222975262623)×(t_SiO₂[λ]−0.142222975262623)−0.00116579711361478).  Equation 12 17: Theacoustic wave device according to claim 4, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (100); in the silicon substrate of which theplane orientation is (100), a directional vector obtained by projectingthe Z_(P)-axis onto a (100) plane of the silicon substrate is k₁₀₀, andan angle between the directional vector k₁₀₀ and a [001] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₀₀; and where the angle α₁₀₀ is Si_psi[deg.], a thickness of thetitanium oxide layer is t_TiO₂[λ], a thickness of the silicon oxidelayer is t_SiO₂[λ], and a thickness of the polysilicon layer ist_Si₂[λ], the Si_psi[deg.], the t_TiO₂[λ], the t_SiO₂[λ], and thet_Si₂[λ] each are a value within a range with which y in equation 13 isless than or equal to about −70:y[deg.]=(−47.9946211404703)+1.21901050350713×(Si_psi[deg.]−19.0566037735849)+4.12041154986452×(t_Si₂[λ]−0.408490566037736)+228.432202102143×(t_TiO₂[λ]−0.0288364779874214)+(−33.1253993677708)×(t_SiO₂[λ]−0.160062893081761)+0.140472008263765×((Si_psi[deg,]−19.0566037735849)×(Si_psi[deg.]−19.0566037735849)−38.1037142518096)+(−5.44594625372052)×((Si_psi[deg.]−19.0566037735849)×(t_Si₂[λ]−0.408490566037736))+114.747133042737×((t_Si₂[λ]−0.408490566037736)×(t_Si₂[λ]−0.408490566037736)−0.0406983505399312)+10.4171695197979×((Si_psi[deg.]−19.0566037735849)×(t_TiO₂[λ]−0.0288364779874214))+(−526.442885320397)×((t_Si₂[λ]−0.408490566037736)×(t_TiO₂[λ]−0.0288364779874214))+(−298.795469471375)×((t_TiO₂[λ]−0.0288364779874214)×(t_TiO₂[λ]−0.0288364779874214)−0.000424904078161465)+(−50.1009078768921)×((Si_psi[deg.]−19.0566037735849)×(t_SiO₂[λ]−0.160062893081761))+1038.08065133921×((t_Si₂[λ])−0.408490566037736)×(t_SiO₂[λ]−0.160062893081761))+(−1286.74436136556)×((t_TiO₂[λ]−0.0288364779874214)×(t_SiO₂[λ]−0.160062893081761))+4158.8148931551×((t_SiO₂[λ]−0.160062893081761)×(t_SiO₂[λ]−0.160062893081761)−0.00134527906332819).  Equation 13 18: Theacoustic wave device according to claim 4, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (110); in the silicon substrate of which theplane orientation is (110), a directional vector obtained by projectingthe Z_(P)-axis onto a (110) plane of the silicon substrate is k₁₁₀, andan angle between the directional vector k₁₁₀ and a [001] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₀; and where the angle α₁₁₀ is Si_psi[deg.], a thickness of thetitanium oxide layer is t_TiO₂[λ], a thickness of the silicon oxidelayer is t_SiO₂[λ], and a thickness of the polysilicon layer ist_Si₂[λ], the Si_psi[deg.], the t_TiO₂[λ], the t_SiO₂[λ], and thet_Si₂[λ] each are a value within a range with which y in equation 14 isless than or equal to about −70:y[deg.]=(−66.0681190864303)+(−0.0323391014318074)×(Si_psi[deg.]−35.9295352323838)+0.997507104337367×(t_Si₂[λ]−0.394527736131933)+155.754971155735×(t_TiO₂[λ]−0.0378860569715143)+(−27.4736558331949)×(t_SiO₂[λ]−0.149887556221888)+0.00791197424152189×((Si_psi[deg.]−35.9295352323838)×(Si_psi[deg.]−35.9295352323838)−256.81962242267)+0.212504000649305×((Si_psi[deg.]−35.9295352323838)×(t_Si₂[λ]−0.394527736131933))+88.6294722534935×((t_Si₂[λ]−0.394527736131933)×(t_Si₂[λ]−0.394527736131933)−0.0392241772666888)+0.636412965393882×((Si_psi[deg.]−35.9295352323838)×(t_TiO₂[λ]−0.0378860569715143))+157.120610191294×((t_Si₂[λ]−0.394527736131933)×(t_TiO₂[λ]−0.0378860569715143))+544.188337615988×((t_TiO₂[λ]−0.0378860569715143)×(t_TiO₂[λ]−0.0378860569715143)−0.000522930045472021)+0.408031229502175×((Si_psi[deg.]−35.9295352323838)×(t_SiO₂[λ]−0.149887556221888))+(−46.0736528123303)×((t_Si₂[λ]−0.394527736131933)×(t_SiO₂[λ]−0.149887556221888))+(−1322.9465191866)×((t_TiO₂[λ]−0.0378860569715143)×(t_SiO₂[λ]−0.149887556221888))+359.098768522305×((t_SiO₂[λ])−0.149887556221888)×(t_SiO₂[λ]−0.149887556221888)−0.00163979245384803).  Equation 14 19: Theacoustic wave device according to claim 4, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (111); in the silicon substrate of which theplane orientation is (111), a directional vector obtained by projectingthe Z_(P)-axis onto a (111) plane of the silicon substrate is k₁₁₁, andan angle between the directional vector k₁₁₁ and a [11-2] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₁; and where the angle α₁₁₁ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the titanium oxide layeris t_TiO₂[λ], a thickness of the silicon oxide layer is t_SiO₂[λ], and athickness of the polysilicon layer is t_Si₂[λ], the Si_psi[deg.], thet_LT[λ], the t_TiO₂[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are a valuewithin a range with which y in equation 15 is less than or equal toabout −70:y[deg.]=(−69.4030815485713)+(−0.269371737613053)×(Si_psi[deg.]−33.8730694980695)+(−4.68577968707475)×(t_Si₂[λ]−0.440745656370656)+176.177168052005×(t_LT[λ]−0.168858590733587)+73.1412385401181×(t_TiO₂[λ]−0.0300916988416992)+(−12.3739066281753)×(t_SiO₂[λ]−0.154983108108108)+0.00777703537774127C((Si_psi[deg.]−33.8730694980695)×(Si_psi[deg.]−33.8730694980695)−508.406668570442)+0.121265989497045×((Si_psi[deg.]−33.8730694980695)×(t_Si₂[λ]−0.440745656370656))+17.8168374568741×((t_Si₂[λ]−0.440745656370656)×(t_Si₂[λ]−0.440745656370656)−0.0493816592475423)+1.34130425597794×((Si_psi[deg.]−33.8730694980695)×(t_LT[λ]−0.168858590733587))+(−5.11032690396319)×((t_Si₂[λ]−0.440745656370656)×(t_LT[λ]−0.168858590733587))+2.48016332864734×((Si_psi[deg.]−33.8730694980695)×(t_TiO₂[λ]−0.0300916988416992))+77.8145877606436×((t_Si₂[λ]−0.440745656370656)×(t_TiO₂[λ]−0.0300916988416992))+2112.87481803881×((t_LT[λ]−0.168858590733587)×(t_TiO₂[λ]−0.0300916988416992))+196.040518466468×((t_TiO₂[λ]−0.0300916988416992)×(t_TiO₂[λ]−0.0300916988416992)−0.000562395066225887)+(−0.969575065396993)×((Si_psi[deg.]−33.8730694980695)×(t_SiO₂[λ]−0.154983108108108))+(−138.70694337489)×((t_Si₂[λ]−0.440745656370656)×(t_SiO₂[λ]−0.154983108108108))+(−1100.04408119143)×((t_LT[λ]−0.168858590733587)×(t_SiO₂[λ]−0.154983108108108))+74.9944030678128×((t_TiO₂[λ]−0.0300916988416992)×(t_SiO₂[λ]−0.154983108108108))+117.812778429437×((t_SiO₂[λ]−0.154983108108108)×(t_SiO₂[λ]−0.154983108108108)−0.00117057162586093).  Equation 15 20: Theacoustic wave device according to claim 4, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (110); in the silicon substrate of which theplane orientation is (110), a directional vector obtained by projectingthe Z_(P)-axis onto a (110) plane of the silicon substrate is k₁₁₀, andan angle between the directional vector k₁₁₀ and a [001] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₀; and where the angle α₁₁₀ is Si_psi[deg.], a thickness of thetitanium oxide layer is t_TiO₂[λ], a thickness of the silicon oxidelayer is t_SiO₂[λ], and a thickness of the polysilicon layer ist_Si₂[λ], the Si_psi[deg.], the t_TiO₂[λ], the t_SiO₂[λ], and thet_Si₂[λ] each are a value within a range with which y in equation 16 isless than or equal to about −80:y[deg.]=(−77.5229944626225)+(−0.0245637365901893)×(Si_psi[deg.]−34.5205479452055)+(−1.18326432300356)×(t_Si₂[λ]−0.408904109589041)+131.275052857081×(t_TiO₂[λ]−0.0160045662100456)+(−18.9167659640434)×(t_SiO₂[λ]−0.150228310502283)+0.00233031321934999×((Si_psi[deg.]−34.5205479452055)×(Si_psi[deg.]−34.5205479452055)−75.9116782385689)+(−0.0809397438263331)×((Si_psi[deg.]−34.5205479452055)×(t_Si₂[λ]−0.408904109589041))+43.1653284334043×((t_Si₂[λ]−0.408904109589041)×(t_Si₂[λ]−0.408904109589041)−0.0169869268780884)+(−0.255179621676294)×((Si_psi[deg.]−34.5205479452055)×(t_TiO₂[λ])−0.0160045662100456))+(−101.438420329361)×((t_Si₂[λ]−0.408904109589041)×(t_TiO₂[λ]−0.0160045662100456))+(−834.553397134957)×((t_TiO₂[λ]−0.0160045662100456)×(t_TiO₂[λ]−0.0160045662100456)−0.000126844728008173)+0.935627393438751×((Si_psi[deg.]−34.5205479452055)×(t_SiO₂[λ])−0.150228310502283))+(−21.1152350576513)×((t_Si₂[λ]−0.408904109589041)×(t_SiO₂[λ]−0.150228310502283))+(−41.8110780477077)×((t_TiO₂[λ]−0.0160045662100456)×(t_SiO₂[λ]−0.150228310502283))+263.939639423742×((t_SiO₂[λ]−0.150228310502283)×(t_SiO₂[λ]−0.150228310502283)−0.00160953691541044).  Equation 16 21: Theacoustic wave device according to claim 4, wherein the piezoelectriclayer is a lithium tantalate layer; the piezoelectric layer hascrystallographic axes (X_(P), Y_(P), Z_(P)); the plane orientation ofthe silicon substrate is (111); in the silicon substrate of which theplane orientation is (111), a directional vector obtained by projectingthe Z_(P)-axis onto a (111) plane of the silicon substrate is k₁₁₁, andan angle between the directional vector k₁₁₁ and a [11-2] direction ofsilicon that is a component of the silicon substrate is an angle ofα₁₁₁; and where the angle α₁₁₁ is Si_psi[deg.], a thickness of thepiezoelectric layer is t_LT[λ], a thickness of the titanium oxide layeris t_TiO₂[λ], a thickness of the silicon oxide layer is t_SiO₂[λ], and athickness of the polysilicon layer is t_Si₂[λ], the Si_psi[deg.], thet_LT[λ], the t_TiO₂[λ], the t_SiO₂[λ], and the t_Si₂[λ] each are a valuewithin a range with which y in equation 17 is less than or equal toabout −80:y[deg.]=(−78.9096176038851)+0.034903516437231×(Si_psi[deg.]−45.8453473132372)+(−2.02533584474356)×(t_Si₂[λ]−0.421068152031454)+81.4363661536651×(t_LT[λ]−0.156225425950199)+71.7903756668229×(t_TiO₂[λ]−0.0334862385321101)+(−3.53261283561548)×(t_SiO₂[λ]−0.149606815203146)+0.00246176329064131×((Si_psi[deg.]−45.8453473132372)×(Si_psi[deg.]−45.8453473132372)−143.191023568758)+(−0.0293712406566626)×((Si_psi[deg.]−45.8453473132372)×(t_Si₂[λ]−0.421068152031454))+0.972512275661977×((t_Si₂[λ]−0.421068152031454)×(t_Si₂[λ]−0.421068152031454)−0.0430997109516307)+0.30260322985253×((Si_psi[deg.]−45.8453473132372)×(t_LT[λ]−0.156225425950199))+(−23.2249863870645)×((t_Si₂[λ]−0.421068152031454)×(t_LT[λ]−0.156225425950199))+0.513496313876766×((Si_psi[deg.]−45.8453473132372)×(t_TiO₂[λ]−0.0334862385321101))+(−143.065209527507)×((t_Si₂[λ]−0.421068152031454)×(t_TiO₂[λ]−0.0334862385321101))+(−324.329178613173)×((t_LT[λ]−0.156225425950199)×(t_TiO₂[λ]−0.0334862385321101))+(−98.4001544132927)×((t_TiO₂[λ]−0.0334862385321101)×(t_TiO₂[λ]−0.0334862385321101)−0.000480867110753061)+(−1.15889670547368)×((Si_psi[deg.]−45.8453473132372)×(t_SiO₂[λ]−0.149606815203146))+(−50.3263112114924)×((t_Si₂[λ]−0.421068152031454)×(t_SiO₂[λ]−0.149606815203146))+(−209.199256641353)×((t_LT[λ]−0.156225425950199)×(t_SiO₂[λ]−0.149606815203146))+90.23187502294813×((t_TiO₂[λ]−0.0334862385321101)×(t_SiO₂[λ]−0.149606815203146))+347.448658314796×((t_SiO₂[λ]−0.149606815203146)×(t_SiO₂[λ]−0.149606815203146)−0.00114008131659363)  Equation 17 22: Theacoustic wave device according to claim 1, wherein the piezoelectriclayer is a lithium niobate layer.